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giant molecular clouds ( gmcs ; @xmath4 ) in the galaxy are highly inhomogeneous : they are often filamentary in appearance , consisting of discrete clumps , or cores ( @xmath5 ) , which contain most of the mass of a cloud ( including any young stars ) and are surrounded by a more diffuse component of predominantly atomic gas ( e.g. , williams , blitz , & stark 1995 ) . indeed , gmcs are clumpy on all scales observed , and are possibly even fractal in nature ( falgarone , phillips , & walker 1991 ) . remarkably enough , however , the gross properties of cloud complexes are rather simply interrelated . total masses , mean densities , and average velocity dispersions vary with sizes ( effective radii ) roughly as @xmath6 , @xmath7 , and @xmath8 ( larson 1981 ; sanders , scoville , & solomon 1985 ; solomon et al . 1987 ) , with uncertainties in the exponents typically of order @xmath90.1 . the seeming universality of these results demands a physical explanation . the relationship between size and linewidth ( which term we use interchangeably with velocity dispersion ) is further interesting because @xmath10 is observed to decrease towards smaller radii _ inside _ gmcs , and within individual dense cores ( larson 1981 ; miesch & bally 1994 ; fuller & myers 1992 ; caselli & myers 1995 ) . since the total linewidths of gmcs are mostly nonthermal ( the clouds are stable against gravitational collapse on the largest scales , but have masses several orders of magnitude above the thermal jeans value , so that their support must come largely from nonmagnetic or , very likely , mhd turbulence ) , this decrease reflects a move towards domination by thermal motions on the smallest scales . we should therefore expect the linewidths of low - mass cores to have a smaller turbulent component than those of high - mass cores . this is indeed the case ; in fact , the velocities nearest the centers of small cores are almost ( to within a few percent ) wholly thermal . however , it also happens ( caselli & myers 1995 ) that the nonthermal velocity dispersion shows _ a stronger dependence on radius in low - mass cores _ ( where @xmath11 ) than in massive ones ( @xmath12 ) . because star formation is localized in the cores of gmcs , their overall structure and this aspect specifically  bears strongly on our understanding of this most fundamental process . the goal of this paper is to find a model for the internal structure of molecular cores ( low- and high - mass both ) which quantitatively matches their observed , internal velocity - dispersion profiles , and is also consistent with global properties ( such as the observed mass - radius - linewidth relations ) of large , self - gravitating clumps and even whole gmcs . our approach is to reduce this problem to the specification of a ( total ) pressure - density relation  an equation of state  that , when used to solve the equation of hydrostatic equilibrium , results in a gas cloud with the required features . it is significant that the linewidth profiles of cores are insensitive to the presence or absence of young stars , and thus may be viewed as one of the prerequisites for star formation ( fuller & myers 1992 ; caselli & myers 1995 ) . this justifies our focus on the structure of purely gaseous clouds . to proceed , we shall resort to a virial - theorem treatment of molecular clouds which idealizes them as spheres of gas in hydrostatic equilibrium and satisfying poisson s equation . this is appropriate enough for dense cores , which in many cases are roughly spheroidal ( probably prolate : myers et al . 1991 ) , or even near spherical ( e.g. , williams et al . 1995 ) , overall . ( although their internal density distributions may not be especially smooth on very small scales , we concern ourselves here with a description of their _ bulk _ structure . ) in addition , observations of very massive cores imply that they are self - gravitating and in approximate `` gravitational virial equilibrium , '' this is not the case for low - mass cores , but these still appear to satisfy the full virial theorem if surface - pressure terms are included ( bertoldi & mckee 1992 ) . finally , even the total masses , radii , and linewidths of entire gmc complexes are generally consistent with virial - equilibrium models of spheres ( e.g. , solomon et al .  1987 ; elmegreen 1989 ) and it has been repeatedly confirmed that whole clouds tend to comply with @xmath14 ( larson 1981 ; solomon et al . 1987 ; myers & goodman 1988b ) .    it should be noted that the correct , average mass - radius and size - linewidth relations can be recovered in a purely scale - free description of gmcs ( e.g. , henriksen 1991 ) . but the distinctly different scalings of velocity dispersion with radius inside low- and high - mass clumps is one indication that molecular clouds are in fact not entirely featureless . other current models for these objects ( such as isothermal spheres or negative - index polytropes ) face similar difficulties in accounting simultaneously for their global properties and their internal structures . magnetic fields are an important presence in regions of star formation ( e.g. , heiles et al . 1993 ) , so we begin in  2 by writing down the virial - theorem ( or bonnor - ebert ) relations between the masses and total linewidths  including turbulent velocities  of magnetized spheres , truncated at radii such that the internal pressure just balances that of a diffuse surrounding medium . we then develop a stability criterion for such clouds which depends only on the assumption that their thermal linewidths ( i.e. , kinetic temperatures ) are invariant . when combined with the circumstantial evidence for equipartition between the kinetic and mean - field magnetic energies in gmcs and massive cores ( myers & goodman 1988a , b ; bertoldi & mckee 1992 ) , this stability criterion leads to mass - radius - linewidth relations between critical - mass objects that agree with the observed scalings ( both the exponents and the coefficients ) among gmcs . our analysis is therefore similar to those of , e.g. , chize ( 1987 ) , fleck ( 1988 ) , and especially elmegreen ( 1989 ) , but ours holds for clouds with an _ arbitrary _ equation of state . in addition , we find that critically stable clouds in magnetic equipartition should all satisfy @xmath15 . we therefore conclude that gmcs and massive cores are approximately at their critical masses and magnetically supercritical , with @xmath16 ( see also mckee 1989 ; bertoldi & mckee 1992 ) . the generality of these results allows for the investigation of essentially any pressure - density relation as a potential description of the interiors of molecular cores , just so long as critically stable configurations are at all possible .    in  3 , we discuss polytropic equations of state : @xmath17 , with @xmath18 any positive number . clearly , if @xmath19 , then the total velocity dispersion ( @xmath20 ) stays constant or decreases with decreasing density ( increasing radius ) inside a cloud . if instead @xmath18 is allowed to be less than 1 ( e.g. , maloney 1988 ) , then @xmath21 increases with radius , as required . however , in our analysis , any such `` negative - index '' polytrope would be unconditionally stable against gravitational collapse ; or equivalently ( but independently of any stability criterion ) , it could not self - consistently satisfy the relation @xmath22 . while this is not a problem for low - mass cores , it is inconsistent with what we know of very large clumps , and gmcs overall . thus , we argue that polytropic pressure - density relations give an incomplete picture of interstellar clouds . a corollary to this is that weakly damped alfvn waves , for which @xmath23 ( mckee & zweibel 1995 ) , can not be invoked as the _ sole _ explanation of nonthermal linewidths in gmcs . section 4 describes what is , in our view , a more suitable alternative . there we consider the possibility that pressure varies only logarithmically with density : @xmath24 . we refer to the resulting gas cloud as a `` pure '' logotrope . this term was introduced by lizano & shu ( 1989 ; see also mckee 1989 ) , who actually added a logarithmic term to an otherwise isothermal equation of state in an attempt to account for turbulent linewidths . such models have also been studied in detail by gehman et al . ( 1996 ) . here , however , we dispense with the explicitly isothermal component , for two reasons : ( 1 ) assuming @xmath25 , with @xmath26 the thermal velocity dispersion and @xmath27 some reference density ( lizano & shu 1989 ) , the observational inference that linewidths should be essentially purely thermal at the centers of cores requires that @xmath28 . but then the internal @xmath20 _ decreases _ with increasing radius . ( 2 ) if now @xmath29 , then for large values of @xmath30 such as those suggested by gehman et al . ( 1996 ) , @xmath31 vanishes for @xmath32 rather near unity . thus , real clouds would have to be almost uniform - density , which again is not observed . these difficulties do not extend to the specific equation of state which we examine . instead , an outwards - increasing velocity - dispersion profile obtains for an equilibrium pure logotrope . such a model can moreover account for the observed mass - radius and size - linewidth relations between gmcs . we also demonstrate that the linewidth measurements in both low- and high - mass cores , from a variety of molecular clouds , are quantitatively reproduced if @xmath3 . the argument makes explicit use of the fact that small cores are _ not _ at their critical masses ( while large ones generally are ) , but that they are still in approximate virial equilibrium when the effects of surface pressures are considered . although the logarithmic @xmath33 relation we advocate is phenomenological , its overall viability , along with the failings of other models , ultimately makes for a useful description of molecular clouds and their cores . interstellar clouds can be viewed as essentially `` pressure - truncated '' bodies of gas . ( this term is meant to imply the existence , not of some radius where a gas cloud suddenly ends , but of one where it `` blends in '' with an ambient medium . ) thus , the boundary of a gmc is set by pressure balance with the surrounding , hot ism . note that even if there is no such balance initially , it will eventually obtain after an overall expansion or contraction of the cloud complex . the extent of a core within a gmc is similarly limited by the pressure of a tenuous interclump medium , as has been argued by bertoldi & mckee ( 1992 ) . the equilibrium structure of pressure - truncated isothermal spheres was first described by ebert ( 1955 ) and bonnor ( 1956 ) ( see also mccrea 1957 ) . their expressions connecting the masses , radii , and linewidths of the spheres follow from the assumption of hydrostatic equilibrium and poisson s equation , and therefore are written in terms of an internal pressure profile and gravitational potential . in appendix a , we re - derive these bonnor - ebert relations , but for clouds satisfying an arbitrary gas equation of state ( eqs . [ [ eq:4a]][[eq:4d ] ] ) . we also provide a connection ( eqs . [ [ eq:13a]][[eq:13 ] ] ) with the more transparent and observationally convenient virial - theorem formulation given by equations ( [ eq:12 ] ) below . as various authors have noted ( chize 1987 ; fleck 1988 ; maloney 1988 ; elmegreen 1989 ) , these relations in either guise provide a framework for an understanding of the standard `` larson s laws '' for gmcs ( @xmath6 and @xmath34 ; larson 1981 ) . the development of appendix a refers specifically to `` nonmagnetic '' clouds , meaning only that no _ ordered _ ( mean ) magnetic field is considered to be present . we work on the assumption that the effects of disordered fields ( mhd turbulence ) can be separately dealt with , in an equivalent hydrostatic problem that makes use of an effective equation of state to describe _ all _ of the contributions to gas pressure as a function of density . still , the effects of mean fields must also be considered in any applications to real interstellar clouds .      in the absence of any analytic models for magnetohydrostatic clouds , we proceed by assuming spherical symmetry and turning to the scalar virial theorem : @xmath35 here the mass - averaged , total one - dimensional velocity dispersion of a cloud with radius @xmath36 is related to its mean pressure and density by @xmath37 so that the kinetic ( or internal ) , mean - field magnetic , and gravitational energies are given by the usual @xmath38 @xmath39 and @xmath40 the parameter @xmath41 is essentially a measure of the non - uniformity of a gas sphere , and as such depends on the equation of state and the truncation radius ( in terms of a fixed scale @xmath42 ; eq .  [ [ eq:13a ] ] ) . however , it is generally of order unity : for a power - law density profile @xmath43 , equation ( [ eq:9 ] ) gives @xmath44 . we expect @xmath45 in a realistic gmc or core , and thus @xmath46 . it is often more useful to work in terms of the virial parameter of bertoldi & mckee ( 1992 ) : @xmath47 this _ observable _ quantity can also be written ( cf . mckee & zweibel 1992 ) as @xmath48 although this form of the virial theorem is appropriate for clouds of any shape , the corrective factors for spheroidal clouds are rather near unity ( bertoldi & mckee 1992 ) , and we still allow only for spherical symmetry here . it is further convenient to distinguish between the virial parameters which would obtain for a cloud with and without a mean magnetic field ; these are related by @xmath49 again , @xmath50 refers specifically to the combination ( [ eq:10 ] ) of observables . on the other hand , @xmath51 applies to the @xmath52 ( no mean field ) counterpart of a given cloud ; it is directly observable only in this special case ( since then @xmath53 ) , but may always be calculated for a given gas equation of state , as outlined in appendix a. once this is known , @xmath50 follows with the specification of a mean - field configuration ( see  2.3 ) . the connection ( [ eq:11 ] ) between @xmath50 and @xmath51 is valid insofar as the ratio @xmath54 does not change drastically upon the `` addition '' of a mean magnetic field to a cloud which is already in hydrostatic equilibrium . this must hold along field lines anyway ( force balance is required in that direction ) , and thus everywhere on the surface of a roughly spherical cloud . defining @xmath55 , such reasoning implies that @xmath56 , so manipulation of the virial theorem gives the following : @xmath57 @xmath58 @xmath59 and @xmath60 these relations , of which only two are independent ( they are contained , for example , in eqs . [ 9 ] and [ 10 ] of elmegreen 1989 ) , are also given by harris & pudritz ( 1994 ) for the specific case of critically stable isothermal spheres ( as defined in  2.2 ; @xmath61 and @xmath62 at a cloud radius @xmath63 ) . as written here , they apply to any generic gas cloud , stable or unstable , isothermal or not . any cloud in hydrostatic equilibrium and satisfying poisson s equation has a central region where the potential , density , and velocity dispersion are very nearly constant with radius . equations ( [ eq:13a ] ) and ( [ eq:13b ] ) show that spheres truncated in or just outside this central region ( i.e. , at radii small enough that @xmath64 ) have @xmath65 , and thus @xmath66 by equations ( [ eq:12]a , b ) . roughly this scaling has been observed for the virial parameters of low - mass , high-@xmath67 cores in several gmcs , leading bertoldi & mckee ( 1992 ) to argue that such clumps can be viewed as truncated spheroids which are essentially in pressure equilibrium with an intracloud medium .      given equations ( [ eq:12 ] ) for the equilibrium structure of pressure - truncated gas spheres , we are in a position to question their stability : under what conditions will they be able to withstand the combined effects of self - gravity and surface pressure , and when will they be unstable to wholesale gravitational collapse ? the answer to this depends , of course , on any boundary conditions attached to a perturbation of the cloud . obviously , the total mass should be unchanged by a contraction or expansion of the entire structure . in addition , following maloney ( 1988 ) , we suppose that the central velocity dispersion remains constant as the cloud radius , or the surface pressure , is varied . this stipulation is meant to reflect the fact that the turbulent linewidth decreases steadily towards smaller scales in cores and in entire gmcs . we therefore identify the central velocity dispersion with the thermal part of the total linewidth : @xmath68 . insisting that this be invariant amounts to recognizing the rough uniformity of kinetic temperatures @xmath69 ( which to first order can be understood as a consequence of the competition between cosmic - ray heating and co cooling ) over a large range of scales in interstellar clouds . the stability criterion that follows ultimately leads to a set of results which self - consistently explain some important observational features of gmcs and massive cores . a gas cloud will be stable against radial perturbations if the derivative @xmath70 ( taken with @xmath71 and @xmath72 held fixed ) is @xmath73 : a slight decrease in the cloud radius then leads to an increase in the pressure just inside its boundary , which in turn leads to reexpansion . appendix b shows that , for any equation of state , this condition is just @xmath74\leq0\ , \label{eq:14}\ ] ] where @xmath75 is the internal density at the edge of the cloud . although this stability criterion has been derived without explicitly considering the effects of mean magnetic fields , we expect that it should not be greatly altered by their inclusion . ( this is again implied by our assumption of approximate spherical symmetry , since eq . [ [ eq:14 ] ] must at least be satisfied along field lines at the boundary of a magnetized cloud . )    depending on the equation of state , there may exist a radius for which a pressure - truncated cloud is marginally stable ( the expression [ [ eq:14 ] ] is just 0 ) , and beyond which it is unstable . it is these critical equilibria , which must have @xmath76 that are of particular interest here . once @xmath51 and @xmath41 are known as functions of radius ( as in appendix a ) , the satisfaction , if possible , of equation ( [ eq:15 ] ) sets the boundary @xmath77 of the cloud . this in turn allows for evaluation of the coefficients in equations ( [ eq:12 ] ) . quite generally , @xmath78 @xmath79 @xmath80 and @xmath81 equation ( [ eq:16]c ) shows that a nonmagnetic cloud ( @xmath53 ) on the verge of gravitational collapse has a mean column density which is fixed by the pressure of the surrounding medium , independently of any gas equation of state .    in general , the virial parameter of a given cloud is sensitive to its internal structure ( through the equation of state ) and its total radius . however , for a _ critically stable _ cloud we always have @xmath82 . since @xmath41 is typically of order ( but slightly greater than ) unity , this implies @xmath83 and @xmath84 . the virial theorem ( [ eq:5 ] ) then becomes @xmath85 if there is equipartition @xmath86 between the magnetic and kinematic energies in such a cloud , then @xmath13 as well ; thus , @xmath87 , and the critical @xmath50 is expected to be of order unity ( cf . eq .  [ [ eq:11 ] ] ; see also elmegreen 1989 and mckee & zweibel 1992 ) .    observations of molecular clouds show that @xmath13 ( or equivalently , @xmath88 ) , and are _ consistent _ with @xmath86 ( myers & goodman 1988a , b ) , although actual magnetic - field measurements are few and uncertain . the most massive cores in gmcs similarly tend to show @xmath22 and @xmath50 near 1 ( this is not the case for low - mass cores , however : e.g. , williams , de geus , & blitz 1994 ; williams et al .  1995 ; see also  4.1 below ) . observations of them are also indicative of magnetic equipartition ( bertoldi & mckee 1992 ) , though again the evidence is rather indirect , and not necessarily conclusive . having said this , it does seem that gmcs and massive cores both satisfy equation ( [ eq:17a ] ) , which would imply that they are at or near their critical masses .    thus arrived at , this conclusion depends on the criterion one adopts for cloud stability , i.e. , it follows from the assumption that the thermal linewidth @xmath71 is held fixed during any radial perturbation ( and from the additional proviso that @xmath86 ) . nevertheless , our analysis  which specifies no equation of state  provides a natural explanation for the fact that so many gmcs and large cores appear to be in simple `` gravitational virial equilibrium '' ( i.e. , @xmath89 ) , even though they are threaded by appreciably strong mean magnetic fields . there is also separate evidence for the criticality of massive molecular cores . for instance , bertoldi & mckee ( 1992 ) , in their study of the cores in four gmcs , argue that the most massive are at least magnetically supercritical , a necessary condition for gravitational instability . more fundamentally , star - forming regions clearly must be susceptible to gravitational collapse ; but in the rosette gmc at least , those cores which are most obviously associated with iras sources are also among the heaviest ( williams et al . as mentioned above , the largest cores in several gmcs also have the smallest virial parameters ( as low as 1 ) , so that this would seem to be a feature of clouds which are close to instability . the observation of a mean @xmath90 for gmcs then implies that they , too , are near some critical mass , and certainly in excess of the nonmagnetic jeans or bonnor - ebert value ( for which @xmath52 implies @xmath91 and @xmath92 ) . mckee ( 1989 ) has further argued that gmcs on the whole are magnetically supercritical , and of course they must be strongly self - gravitating in order to be molecular at all ( e.g. , elmegreen 1985 ) .    on a related note , mckee ( 1989 ) points out that gmcs must generally be near criticality because they show a @xmath93 which is typically an order of magnitude larger than the total ( thermal plus turbulent ) pressure in the hot ism . given our stability criterion , equation ( [ eq:16]d ) shows that self - gravity can supply a maximal pressure enhancement @xmath94 of a spherical cloud over its surrounding medium , and this only for a critical - mass body . even putting the nonsphericity of gmcs aside , however , it is important to note that the analysis here speaks only to the molecular , self - gravitating parts of gmcs , and not to their diffuse , low-@xmath95 components . elmegreen ( 1989 ) has shown that the weight of these atomic `` envelopes '' can easily increase the pressure at the boundaries of the molecular parts of a cloud complex by a factor of 5 or more above the value in the ism at large ; overall , then , @xmath96 . if gmcs and their most massive cores are indeed critical - mass objects , then they must all have the same dimensionless radii ( although the physical scale @xmath42 will generally vary ) , and the same virial parameters . aside from possible variations in @xmath97 , which are discussed in detail by elmegreen ( 1989 ) , this causes the coefficients in equations ( [ eq:16 ] ) or ( [ eq:12 ] ) to be roughly constant , and allows for well defined mass - radius - linewidth relations between clouds . moreover , we have argued that @xmath98 and , in the event of magnetic equipartition , @xmath99 for critical clouds , regardless of the underlying equation of state . the coefficients in the @xmath100 scalings are then independent of this detail , and the average properties of gmcs can shed no light on their internal structure . this both explains why critical , magnetized isothermal - sphere models are successful in quantitatively accounting for the observed scalings ( with @xmath101 and @xmath99 : elmegreen 1989 ; harris & pudritz 1994 ) , and implies that the same agreement with the data comes with _ any _ model which provides for the existence of a critical mass ( see eqs . [ [ eq:411 ] ] below ) . finally , equations ( [ eq:16 ] ) as written would suggest that all gmcs must be under similar surface pressures @xmath97 if , for example , @xmath102 is to be roughly the same among them . other authors ( e.g. , mckee 1989 ; mouschovias 1987 ; myers & goodman 1988b ) have argued instead that either @xmath102 itself ( and hence @xmath93 ) , or the mean field strength @xmath103 , is the more fundamentally invariant attribute of gmcs . if so , then @xmath97 could be eliminated from the critical bonnor - ebert relations in favor of any of these quantities , and our approach does not preclude the others .      consider a cloud of radius @xmath36 , threaded by a mean magnetic field approximated as uniform and of magnitude @xmath103 . outside the cloud , let @xmath104 fall off as @xmath105 to a radius @xmath106 , where it matches onto an ambient , uniform field of strength @xmath107 ( nakano 1984 ) . conservation of flux ( or continuity of the normal component of * b * across the boundary of the cloud ) demands @xmath108 , and evaluating equation ( [ eq:8 ] ) at the surface @xmath109 gives @xmath110 defining @xmath111 and @xmath112 , we have @xmath113 so that equation ( [ eq:11 ] ) gives @xmath114 with @xmath115 g and @xmath116 g for gmcs ( myers & goodman 1988b ) , flux conservation gives @xmath117 ; and @xmath118 implies @xmath119 . thus , @xmath88 when @xmath83 , as expected .    some indication of the reliability of equations ( [ eq:16 ] ) and ( [ eq:17 ] ) , and of the approximations leading to them , can be had by comparing the critical masses they predict for magnetized , isothermal spheres ( @xmath62 ) with those obtained from self - consistent , axisymmetric numerical calculations . in particular , the mass @xmath120 which separates magnetically sub- and supercritical clouds , and for which @xmath121 , is given by equation ( [ eq:17 ] ) as @xmath122 , only a 50% overestimate of the exact result @xmath123 ( mouschovias & spitzer 1976 ; tomisaka , ikeuchi , & nakamura 1988 ) . further , for @xmath124 , the critical masses we obtain by using equation ( [ eq:11 ] ) in ( [ eq:16]a ) lie within a factor 2 of those found by tomisaka et al . ( 1988 ; see their eq .  [ 4.7 ] ) . ( in fact , our formula is more accurate in the weak - field limit because it approaches the correct @xmath125 for the nonmagnetic , @xmath126 isothermal sphere . ) and for the equipartition @xmath127 seen in gmcs , setting @xmath90 in equation ( [ eq:17 ] ) implies @xmath128 . this is roughly 1.6 times our ( approximate ) @xmath120 , and 2.4 times the exact value , which level of agreement is quite acceptable . in any case , we are led to expect that critical - mass gmcs and cores are strongly magnetically supercritical , with @xmath129  a result which has also been argued by mckee ( 1989 ) and bertoldi & mckee ( 1992 ) . the outwards increase of linewidth both within giant molecular clouds as a whole ( larson 1981 ; miesch & bally 1994 ) and within individual dense cores ( fuller & myers 1992 ; caselli & myers 1995 ) , immediately suggests the class of negative - index polytropes as possible models for these structures . that is , if @xmath130 , then for a polytropic index @xmath131 we have @xmath132 increasing for decreasing @xmath133 , as required . such models have been studied by , e.g. , viala & horedt ( 1974 ) and maloney ( 1988 ) , and we derive the generalized bonnor - ebert relations for them in appendix c. although there are various physical arguments to support their use ( e.g. , shu et al . 1972 ; de jong , dalgarno , & boland 1980 ; mckee & zweibel 1995 ) , these polytropes turn out to be unlikely descriptions of real gmcs ( and of course , a positive polytropic index is undesirable because it is inconsistent with a velocity dispersion that increases with radius ) .    here we define @xmath134 , so that @xmath135 and @xmath136 for @xmath131 . from appendix c , a nonmagnetic , negative - index polytrope which is truncated anywhere outside of its constant - density central region ( inside which , @xmath137 and @xmath51 increases without bound towards @xmath138 ) will then satisfy @xmath139 thus , an @xmath136 polytrope has @xmath140 ; and since its density profile is @xmath141 , with @xmath142 everywhere for any @xmath143 ( appendix c ) , we also find @xmath144 . according to equation ( [ eq:14 ] ) , then , _ a truncated , negative - index polytrope will always be stable , and never critically so _ ( as was also noted by maloney 1988 ) . ultimately , the same steady increase of linewidth with radius which would recommend the polytropic equation of state in the first place also proves to be its undoing : the internal pressure gradient which results is so shallow as to stabilize a cloud under any external pressure . the concept of a critical mass is then irrelevant for these models , which casts doubt on their utility in describing real gmcs or high - mass cores . any additional support from a mean magnetic field in the cloud obviously serves only to exacerbate this problem . again , this result follows to some extent from the constraint that the thermal linewidth @xmath71 be fixed during a perturbation of the cloud . by contrast , both viala & horedt ( 1974 ) and chize ( 1987 ) consider the possibility that the constant of proportionality in the relation @xmath135 is invariant , and find that polytropes can become unstable for certain truncation radii . still , our approach is closely related to a point which is _ independent of any rule for cloud stability _ : the minimum value of the nonmagnetic virial parameter for a truncated polytrope is ( eq .  [ [ eq : b7 ] ] ) @xmath145 as usual , this is a lower limit because @xmath51 can be very large indeed if the polytrope is truncated at a very small radius . in the event of equipartition @xmath146 between kinematic and magnetic energies , equation ( [ eq:11 ] ) and the identity @xmath147 lead to @xmath148 in which case equation ( [ eq : b11 ] ) and @xmath136 finally imply that @xmath149 thus , regardless of whether there exist any unstable modes for truncated polytropes , the virial parameters of such clouds are significantly larger than the @xmath99 ( @xmath13 ) seen in gmcs ( myers & goodman 1988b ) and in the most massive molecular cores ( bertoldi & mckee 1992 ; williams et al .  1994 , 1995 ) , even if dynamically significant mean magnetic fields are allowed to be present . it is a property of our specific stability criterion that this fact implies the absence of critically stable equilibria . the virial parameters of magnetized polytropes could be reduced to @xmath99 , for any @xmath143 , if the mean field were such that @xmath150 . in this case , however , using equation ( [ eq : b11 ] ) for @xmath54 in the virial theorem ( [ eq:5 ] ) leads to @xmath151 for @xmath152 , which is difficult to reconcile with the rather higher degree of magnetic supercriticality that is observationally inferred for gmcs and massive cores ( @xmath153 , and @xmath154 ; see   2.2 , 2.3 ) . one consequence of all of this is that the nonthermal linewidths in gmcs , @xmath155 , can not be attributed entirely to the pressure of weakly damped alfvn waves , for which @xmath156 ( @xmath157 , or @xmath158 ; mckee & zweibel 1995 ) , and thus @xmath159 under magnetic equipartition . it seems almost certain that alfvn waves do play a significant role in the support of gmcs and cores ( e.g. , arons & max 1975 ; pudritz 1990 ) ; but , as mckee & zweibel ( 1995 ) also note , they can not be uniquely responsible for their large - scale stability . in this context , we note that the size - linewidth relation between clouds can be expressed in terms of a mean magnetic field strength , as in , e.g. , myers & goodman ( 1988a ) . specifically , equation ( [ eq:16]d ) can be used to write ( [ eq:16]b ) in terms of @xmath93 rather than @xmath97 , and the definition of @xmath160 (  2.3 ) relates @xmath93 to @xmath103 . then , with a mean mass per particle @xmath161 and a kinetic temperature @xmath162 k , we have @xmath163 the equality is not quite exact here , because the scalings with @xmath103 and @xmath36 strictly apply to the total @xmath164 . still , this relation is accurate in the typical case , @xmath165 ; and for @xmath166 , it is consistent with available data ( see myers & goodman 1988a ) . thus , although mouschovias & psaltis ( 1995 ) argue for an interpretation of this result in terms of alfvn waves , we see here that it is independent of _ any _ assumptions on the physical origin of the nonthermal motions in gmcs ( in principle , they need not even derive from magnetic fields ) .    finally , it is clear that any simple `` mixing '' of two polytropes with different indices will still preclude the existence of a critical ( or low-@xmath50 ) cloud ; even the superposition of an isothermal part , i.e. , @xmath167 , only admits one if it is essentially isothermal anyway ( @xmath168 ) . thus , we now turn to a different equation of state , in which the gas pressure varies only logarithmically with density . many theoretical models of star - forming clouds employ the singular isothermal sphere , in which @xmath169 . however , there is some evidence that regions are concentrated towards the co centroids of their parent gmcs such that their three - dimensional number density is most consistent with an @xmath170 fall - off ( waller et al . 1987 ; scoville et al . this suggests the possibility that , in a heavily smoothed , average sense , the internal density structure of molecular clouds is essentially @xmath171 ( see also solomon et al . further , extinction measurements indicate @xmath171 or so in the outer parts of cores as well ( cernicharo , bachiller , & duvert 1985 ; stwe 1990 ) . lizano & shu ( 1989 ; see also gehman et al . 1996 ) introduced the so - called logotropic equation of state for gmcs , @xmath172 , with @xmath173 and @xmath174 . if the central linewidth of a cloud is to be entirely thermal in origin , the reference density in @xmath175 must be @xmath28 . however , @xmath132 then _ decreases _ with radius , which is incompatible with the observations . we suggest that a more complete description of gmcs and cores is given instead by a `` pure '' logotrope : @xmath176\ , \label{eq:41}\ ] ] with @xmath177 a parameter to be adjusted . the assumption here is that any nonthermal motions , which presumably arise from mhd turbulence , add to the thermal pressure such that both are fully accounted for by the equation of state ( [ eq:41 ] ) . this relation is shown schematically in fig . [ fig0 ] , along with an isothermal - sphere combination and the polytropic @xmath23 for alfvn waves . in the rest of our discussion , equation ( [ eq:41 ] ) is referred to simply as a logotrope , with the understanding that no further consideration is given to any mixed - isothermal version .    given equation ( [ eq:41 ] ) , the equation of hydrostatic equilibrium ( [ eq:1 ] ) integrates to @xmath178 where @xmath179 is a dimensionless gravitational potential . substitution of this expression into poisson s equation ( [ eq:2 ] ) shows the existence of a singular solution , @xmath180 which describes the density profile of the bounded ( finite @xmath181 ) solution at large radii . ( the scale radius @xmath182^{1/2}$ ] . ) in spite of such a slowly decreasing internal density , a logotrope has only a finite extent , since for any positive @xmath2 the pressure will eventually vanish . assuming that @xmath2 is _ small _ enough for the singular density profile to apply at the edge of the cloud , we have @xmath183 the velocity dispersion , @xmath20 , increases with radius inside the cloud until it reaches a maximum of @xmath184 at @xmath185 , beyond which it falls again ( to 0 , at @xmath186 ) . a logotrope truncated by the pressure @xmath97 of an external medium has precisely one critical mode for any given value of @xmath2 . as discussed in appendix b , if a cloud is truncated at a very small radius @xmath187 , it will always be stable . but in the power - law part of the logotrope , equations ( [ eq:42 ] ) and ( [ eq:43 ] ) show that @xmath188 , so once @xmath187 is so large that @xmath189 , the cloud is unstable ( cf . [ [ eq : a5a ] ] ) . thus , @xmath190 where we have again assumed that @xmath2 is relatively small and the singular solution ( [ eq:43 ] ) holds at @xmath191 . ( this expression shows that @xmath10 achieves its maximum _ inside _ @xmath191 , and the surface of the critical logotrope is cooler than its interior . mckee [ 1989 ] has suggested that such behavior might arise from the radiation of alfvn waves into the ambient medium . ) now , evaluating the pressure ( [ eq:41 ] ) and density ( [ eq:43 ] ) at the radius ( [ eq:45 ] ) yields , for any critically stable logotrope , @xmath192 evidently , self - consistency in the use of equation ( [ eq:43 ] ) here requires @xmath193 . equations ( [ eq:4d ] ) and ( [ eq:6 ] ) can then be used to find @xmath194 independently of @xmath2 . a numerical solution for our logotrope with @xmath195 ( which we justify shortly ) yields the internal density , pressure , and velocity - dispersion profiles shown in fig . [ fig1 ] . equations ( [ eq:44 ] ) and ( [ eq:45 ] ) are valid here , as an @xmath170 density profile is realized well before the cloud ends ( the vertical line in all four panels of the figure marks the radius @xmath191 ) . it is the steep pressure gradient near @xmath186 which sets the logotrope apart from the negative - index polytropes and allows for a critical cloud with a small virial parameter . the _ total _ velocity dispersion @xmath21 in fig . [ fig1 ] is roughly constant near the center of the cloud ; and its rise with radius further on is consistent with @xmath196 , which is essentially the scaling originally found for gmcs by larson ( 1981 ) . identifying the central dispersion @xmath71 with the thermal part of the linewidth ( so that thermal motions dominate on the smallest scales , as is observed ) , the nonthermal contribution to cloud support is @xmath197 , which is also shown in fig . [ fig1 ] . at small to moderate radii , @xmath198 naturally rises more steeply than the total @xmath10 , while at larger @xmath187 the two are more comparable in magnitude . roughly , then , our model has @xmath199 over some range in radius before it flattens to @xmath200 and eventually turns over ( the maximum @xmath201 is just @xmath202 , and occurs at @xmath203 ) . it is found that @xmath11 within gmcs ( e.g. , myers & goodman 1988b ) , and low - mass molecular cores show the same general trend ; high - mass cores are more suggestive of @xmath204 ( fuller & myers 1992 ; caselli & myers 1995 ) . a logotropic equation of state both for individual cores and for entire cloud complexes is consistent with these observations , provided that low - mass cores can be viewed as spheres which are pressure - truncated at radii @xmath205 . we discuss this point further in  4.1 below . as always , though , no matter how @xmath10 varies with @xmath206 internally , the _ average _ relation @xmath207 still holds for critical - mass clouds , i.e. , for gmcs .    a handle on an appropriate value of the parameter @xmath2 for gmcs as a whole , can be gotten by reinventing them as smoothed - out spheres with a number of distinct overdense regions ( the cores ) sprinkled throughout . the observed contrasts in mean density and velocity dispersion between gmcs and cores can then be related to the @xmath2 of a model logotrope . harris & pudritz ( 1994 ; see their table 2 ) list each of @xmath208 , @xmath164 , and @xmath102 for a gmc of mass @xmath209 and a core with @xmath210 . ( such values are `` typical '' in the sense that , _ by mass _ , half of all gmcs [ cores ] are larger than @xmath211 [ @xmath212 . ) defining @xmath213 , @xmath214 , and @xmath215 as the core - to - gmc ratios of mean volume densities , linewidths , and column densities , these data show that @xmath216 cores with @xmath217 are among the most massive observed , and are likely near critical . if they are also logotropes , then their @xmath218 , etc . , should be roughly given by equation ( [ eq:48 ] ) , regardless of whether @xmath219 . such large cores might further be expected to lie in very dense , cold , highly pressured regions of their parent clouds . then the pressure @xmath97 at the surface of a core is comparable to the gmc s @xmath220 , and similarly for the density and velocity dispersion . these assumptions , together with the relation @xmath221 ( eq .  [ [ eq:16]c ] ) , result in @xmath222 comparison of equations ( [ eq:46 ] ) and ( [ eq:47 ] ) then shows that the observed @xmath213 , @xmath223 , and @xmath224 ( only two of which are actually independent ) are realized in a logotropic gmc with @xmath225    although this procedure is highly idealized , it is perhaps telling that an @xmath2 exists at all which gives at once the correct @xmath226 and @xmath214 . the logotrope appears to be the simplest barotropic equation of state which can account for these ratios _ and _ the small virial parameters ( or the critical equilibrium ) of gmcs and massive cores . for example , if gmcs and large cores were both @xmath157 polytropes truncated at @xmath227 , then equation ( [ eq:46 ] ) could be roughly satisfied , but they would also have a large @xmath228 , and hence @xmath229 for magnetic equipartition ( again , @xmath150 is required to reduce @xmath50 to unity for a negative - index polytrope , but this is not wholly satisfactory for other reasons ; see  3 ) . alternatively , adding an explicit @xmath230 term to equation ( [ eq:41 ] ) destroys the _ simultaneous _ correspondence between @xmath213 and @xmath223 for critical clouds .    to this point , we have not accounted for any mean magnetic fields in gmcs and cores . however , these should have no effect on our estimate of @xmath231 insofar as very massive clumps have the same @xmath88 as their parent clouds . more important is the assumption that the cores here lie at the rare , highest - pressure peaks of cloud complexes ; if instead they are found in less extreme regions , then equation ( [ eq:410 ] ) is an upper limit to @xmath231 . even so , more direct estimates of @xmath231 are in fairly good agreement with the representative value quoted above . for example , equations ( [ eq:47 ] ) and ( [ eq:48 ] ) can be combined to obtain an expression for @xmath232 ; comparison with observed values of the total @xmath164 and the thermal @xmath71 in gmcs then suggests @xmath233 .    on a final note , since @xmath171 at the edges of such logotropes , we have @xmath234 . with the ratio of mean kinematic and magnetic pressures @xmath119 , equations ( [ eq:15 ] ) and ( [ eq:18 ] ) then give for the critical virial parameters , @xmath235 these are near 2 and 1 , as expected from general arguments , and not far from the values 2.054 and 1.15 for a bonnor - ebert isothermal sphere . as alluded to earlier (  2.2 ) , the mass - radius and size - linewidth relations among real gmcs can therefore be understood in terms of this model : equations ( [ eq:16]b , c ) become @xmath236 and @xmath237 which agree well with observations ( cf . elmegreen 1989 ) . here @xmath97 is the surface pressure on the _ molecular part _ of the gmc , while @xmath238 is the total pressure of the hot , intercloud medium . the ratio of these two is typically around @xmath239 , due to the weight placed on a gmc by its uv - shielding atomic layer ( elmegreen 1989 ; also  2.2 above ) .      as a class , dense molecular cores seem somewhat more heterogeneous than their parent gmcs . first , many studies ( carr 1987 ; loren 1989 ; stutzki & gsten 1990 ; lada , bally , & stark 1991 ) show that the size - linewidth relation between cores can be significantly weaker than the virial , @xmath8 scaling among gmcs , and in some cases is difficult to discern at all . second , in the l1630 gmc at least , the size - linewidth relation is considerably better defined for clumps which stand out more strongly against the interclump medium ( see the comparison in fig . 13 of lada et al .  between their `` 5@xmath10 '' and `` 3@xmath10 '' cores ) . and third , the virial parameters @xmath50 of gmc cores are not restricted to values near 1 , but rather can range as high as @xmath240100 ( bertoldi & mckee 1992 ; williams et al . 1994 , 1995 ) . this last point is particularly significant because the virial parameters of cores in a number of different clouds are seen to correlate with their size : low mass goes along with rather large @xmath50 , and high mass with more moderate @xmath88 . thus , as bertoldi & mckee ( 1992 ) argue , the smallest clumps in gmcs are far removed from gravitational instability ( surface pressure is more important than self - gravity in their confinement ) , while the largest are actually near criticality ( strongly self - gravitating ; see also  2.2 above ) . equivalently , a range in the masses of cores can be viewed , at least roughly , as a range in the dimensionless radii @xmath187  small for low mass , and near @xmath191 for high mass  at which they are truncated by the ambient pressure in a gmc ( virial parameters generally decrease with @xmath187 for any equation of state ; appendix a ) . this notion could account for the rather confused size - linewidth relation _ between _ cores ( if @xmath187 does not have a fixed value , then neither do @xmath51 and @xmath50 in eq . [ [ eq:12]b ] ) , and ultimately must also have consequences for the interpretation of data on the radial variation of velocity dispersion _ inside _ cores .    if indeed very low - mass cores differ from very high - mass ones mainly in being truncated at @xmath241 , then the former must be less centrally condensed , have ratios @xmath242 nearer unity , and be generally less distinguishable from the intercore ( gmc ) gas than the latter . this results in the scale radius @xmath243 being _ larger _ for less massive clumps . to see this , compare the @xmath42 of a low - mass core , truncated at @xmath187 small enough that @xmath64 , with the @xmath42 of a near - critical , high - mass core . on average , any two cores should be under similar surface pressures @xmath97 ( this essentially being set by the internal @xmath93 of a parent gmc ) , so that @xmath244\   \simeq\ \sqrt{{a\over{4}}}\   \left[{{\sigma_c^2{\rm{(high)}}}\over{\sigma_c^2{\rm{(low)}}}}\right]\ , \label{eq:412}\ ] ] where the second step follows from equation ( [ eq:47 ] ) if a logotropic equation of state applies . thus , for @xmath71 not too widely different between the two cores , and for @xmath2 small enough , the ratio ( [ eq:412 ] ) will be @xmath245 . the same _ physical _ radius @xmath206 in high- and low - mass cores then corresponds to respectively larger and smaller @xmath246 . this result , which just reflects the fact that more strongly self - gravitating structures are more centrally concentrated , must be considered when attempting to match any model to any observations . returning now to the issue of internal velocity - dispersion profiles , fuller & myers ( 1992 ; hereafter fm ) and caselli & myers ( 1995 ; hereafter cm ) have compiled linewidth measurements in at least three different molecular lines for each of 14 low - mass and 24 massive cores . the resulting @xmath10 vs.  @xmath206 profiles indicate that , over similar ranges @xmath247 pc in all the clumps , and independently of whether or not stars are present within them , the velocity dispersion rises more steeply with radius in low - mass cores than in high - mass ones . in particular , nonthermal linewidths in the former are consistent with @xmath248 ; in the latter , @xmath249 ( cm ) . in our view , this situation is a direct result of the effect summarized by equation ( [ eq:412 ] ) . that is , we consider the low - mass cores of fm to be eminently stable logotropes truncated at small @xmath187 , while the high - mass clumps of cm are near critical stability and hence have smaller @xmath42 . the two surveys target similar @xmath206 within each type of core , so the massive - core data apply to larger dimensionless @xmath187 , and must sample the shallower parts of velocity - dispersion profiles like those in fig .  [ fig1 ] .    to actually confront the predictions of our logotropic equation of state with the observations of fm and cm , we again identify the ( model ) central velocity dispersion with the ( observed ) thermal linewidth : @xmath250 , where @xmath251 is taken from cm , and @xmath252 . cm assume @xmath162 k for all the low - mass cores , and we further assume them all to have @xmath64 ; they should therefore have a common @xmath42 , which may be easily estimated . five of the fm cores are observed at radii @xmath253 such that @xmath254 ; among these , @xmath255 pc ( see table 3 of cm ) . the related quantity @xmath256 in a model logotrope is just that point where @xmath257 . if , for example , @xmath258 in equation ( [ eq:41 ] ) , we find @xmath259 , and thus @xmath260 this result is then used in equation ( [ eq:412 ] ) , along with @xmath261 and @xmath258 , to give @xmath262 where @xmath263 for the massive cores ( cm ) . the data may now be appropriately scaled and compared to theoretical @xmath264 and @xmath265 curves . figure [ fig2 ] is the result of such a comparison with three logotropes of different @xmath2 . the good overall agreement between the models and data suggests that gmc cores are well described by @xmath266 which is encouragingly close to @xmath231 as given in equation ( [ eq:410 ] ) above . in table [ tbl1 ] we list the radii and other properties ( as calculated from eqs . [ [ eq:44 ] ] to [ [ eq:47 ] ] above ) of critical logotropes with @xmath2 near 0.2 . some of the scatter in fig . [ fig2 ] could arise if @xmath2 does not have a common value for all cores , and any embedded stars could further complicate the situation . ( notably , fm and cm both find that starless cores alone show a tighter  but not different  linewidth - radius relation than do cores with stars , suggesting that the basic form seen here is one of the initial conditions for star formation . ) however , some of the scatter is surely due to the fact that the cores in this sample do not reside in a single gmc , but come from diverse environments . ( for instance , the low - mass cores here tend to be found in dark clouds of mass @xmath267 , and the high - mass ones in somehat larger gmcs . ) thus , there is no guarantee that every low - mass core observes @xmath64 , or that every high - mass core is at its critical mass . it is the large number of cores available here which allows the mean trend in fig . [ fig2 ] , and the implied structural dichotomy between small and large clumps , to emerge .    as with our earlier consideration of core - to - gmc density and linewidth ratios , a fit of similar quality to the data in fig . [ fig2 ] can be obtained with the model velocity - dispersion profile of an @xmath157 polytrope which has a scale @xmath42 about a factor of two smaller than that used here . as we have stressed , however , there are other difficulties with the application of polytropic equations of state to gmcs and cores .    in summary , the approach we have taken self - consistently allows for purely thermal motions to dominate on the smallest scales in cores ( and in gmcs ) ; for an internal density structure ( @xmath171 in the outer parts of cores , and much shallower near their centers ) which is consistent with at least some observations ( cernicharo et al . 1985 ; stwe 1990 ; andr , ward - thompson , & barsony 1993 ) ; and for a unified treatment of low- and high - mass cores . this is in contrast to the models of fm and cm ( also myers & fuller 1992 ) , which assume two distinct components of isothermal and nonthermal gas in cores and predict a @xmath169 singularity at their centers . it should also be noted that the assumption @xmath64 in low - mass cores still allows for these noncritical clumps to show a fair degree of central concentration . for example , an @xmath258 logotrope which is truncated at @xmath268 has @xmath269 , but @xmath270 ; thanks to the relatively weak dependence of @xmath31 on @xmath133 , we do not require , nor even expect , that all low - mass cores be uniform - density spheres . our model is therefore consistent with the significantly non - uniform density profiles of low - mass cores ( @xmath271 ) in the rosette gmc ( williams et al . we have generalized the classic bonnor - ebert relations for isothermal spheres , to give expressions for the masses , radii , and total velocity dispersions of magnetized , pressure - truncated gas clouds with any internal pressure - density relation . the analysis combines an exact approach , based on solving the equation of hydrostatic equilibrium , with a virial - theorem treatment to incorporate mean magnetic fields into the clouds . a stability criterion has been developed that relies only on the assumption of an invariant central velocity dispersion @xmath71 , and is independent of the gas equation of state . clouds which are just _ critically _ stable under this condition have mass - radius and size - linewidth relations which are effectively oblivious to their internal structure , i.e. , to their equation of state .    these results have been applied to molecular cloud complexes and dense cores , leading to three main conclusions :    \(1 ) in spite of their nonspherical geometry , the gross features of gmcs are consistent with those of critical - mass clouds in our analysis . ( in the context of gmcs and cores , we identify the central velocity dispersion with the _ thermal _ part of the total [ thermal plus turbulent ] linewidth . ) _ independently of the exact equation of state _ , if equipartition between magnetic and kinetic energies obtains in a critically stable cloud , then it will appear to be in simple gravitational equilibrium , @xmath89 ( equivalently , it will have a virial parameter @xmath272 ) . these are observed features of real gmcs and their most massive cores . in addition , any of our critical - mass spheres obey the same mass - radius and size - linewidth relations which hold among real gmcs . all of this implies that gmcs , and large clumps , are near criticality . \(2 ) no polytropic ( power - law ) pressure - density relation can fully characterize gmcs or cores . a positive polytropic index leads to a velocity dispersion which decreases outwards within a cloud , opposite to what is seen both in individual cores and in entire cloud complexes . alternatively , a negative index ( i.e. , @xmath17 with @xmath273 ) gives a linewidth which increases with radius , as required . however , such clouds do not show virial parameters near unity unless their mean fields are so strong that their magnetic and gravitational energies are comparable in magnitude , and this is again in conflict with observations of gmcs and large cores . given our stability criterion , an equivalent statement is that pressure - truncated polytropes are unconditionally stable against gravitational collapse . since weakly damped alfvn waves satisfy @xmath23 ( mckee & zweibel 1995 ) , this shows explicitly that they can not fully explain the observed nonthermal support in molecular clouds . \(3 ) the most successful model for the internal structure of molecular cores , and one which is also consistent with global properties of gmcs , is a `` pure logotrope : '' @xmath24 . this relation is meant to account for _ all _ contributions to the total gas pressure , including the effects of disordered magnetic fields ( mhd turbulence ) . once it is recognized that low - mass cores are far below their critical masses ( though still in virial equilibrium ; they are essentially pressure - confined ) , and that high - mass cores are near criticality , the internal velocity - dispersion profiles of clumps from a variety of environments are seen to be consistent with a logotropic model with @xmath3 . a similar value of @xmath2 can also explain characteristic core - to - gmc ratios of mean densities and linewidths , and any @xmath2 is consistent with the observed size - linewidth relation between gmcs because the logotropic equation of state allows for critical equilibria . finally , the equilibrium density profile of a logotrope has @xmath171 , outside of a constant - density central region . there is some observational support to be found for this prediction , in gmcs and cores both . we would like to thank phil myers for his comments on an earlier version of this paper , and charles curry for valuable discussions . r.e.p .  also acknowledges conversations with paola caselli and gary fuller . this work was supported in part by the natural sciences and engineering research council of canada . for a spherical cloud in hydrostatic equilibrium , with kinematic ( thermal plus turbulent ) pressure @xmath274 ( so that @xmath10 is the one - dimensional velocity dispersion ) and a self - gravitational potential @xmath275 , we define @xmath276 where a subscript @xmath277 denotes evaluation of a quantity at the cloud center . the characteristic scale of a cloud is set by @xmath42 ; the factor of 9 in its definition identifies it with the projected half - power radius of an isothermal sphere ( e.g. , binney & tremaine 1987 ) . the equation of hydrostatic equilibrium then reads @xmath278 and poisson s equation becomes @xmath279 ( note that @xmath280 at @xmath281 , and @xmath282 for @xmath283 . ) the mass enclosed within radius @xmath187 is given by @xmath284 so that , with the help of equation ( [ eq:2 ] ) , @xmath285    following ebert ( 1955 ) and bonnor ( 1956 ) , the surface of the cloud is defined by that radius @xmath187 at which the internal pressure @xmath286 just equals a confining pressure @xmath97 due to a surrounding medium of negligible gravity . equation ( [ eq:3 ] ) , the definition of @xmath42 , and the identity @xmath287 then yield , for any equation of state relating @xmath31 and @xmath133 , @xmath288 the physical radius of the cloud is @xmath289 : @xmath290 these relations combine to give @xmath291 and @xmath292 given an equation of state , the integration of equations ( [ eq:1 ] ) and ( [ eq:2 ] ) determines @xmath293 and @xmath242 at any truncation radius @xmath187 . this fixes the coefficients in equations ( [ eq:4a])([eq:4d ] ) and affords mass - radius and size - linewidth relations for pressure - truncated clouds . ( note that the scalings in these relations  @xmath294 , @xmath295 , etc . are _ global _ ones , and do not necessarily reflect the radial dependence of any quantity _ inside _ a cloud . ) the definition of the non - uniformity parameter @xmath41 ( eq .  [ [ eq:9 ] ] ) allows it to be found as a function of radius : @xmath296 and equation ( [ eq:3 ] ) implies that the virial parameter @xmath297 for nonmagnetic clouds ( i.e. , @xmath53 ) is just @xmath298 which tends to be a nonincreasing function of @xmath187 . alternatively , equations ( [ eq:4d ] ) and ( [ eq:12]d ) give @xmath299 these results can be used to rewrite equations ( [ eq:4a])([eq:4d ] ) in terms of @xmath51 , @xmath41 , and @xmath164 ; equations ( [ eq:12 ] ) ( with @xmath300 ) are then obtained . also , once @xmath51 is known as a function of radius in any cloud , the observable @xmath50 for a given magnetic field configuration may be calculated ( e.g. ,  2.3 ) . we begin by recalling the definition of the scale radius @xmath42 from equation ( [ eq:3 ] ) : @xmath301 and rearranging equation ( [ eq:4a ] ) : @xmath302 the stability of a cloud truncated at radius @xmath36 by the external pressure @xmath97 is determined by the sign of the derivative @xmath303 : this is negative for a stable equilibrium , 0 for the `` critical '' cloud , and positive for instability . we now apply a perturbation @xmath304 , while keeping the mass @xmath72 and central ( i.e. , thermal ) velocity dispersion @xmath71 of the cloud fixed , so that @xmath305 then @xmath289 and equation ( [ eq : a1 ] ) give us @xmath306^{-1 } \nonumber \\   & = & { 1\over{r_0}}\left[1-\xi\left(\xi^2\dpsi\right)^{-1 } { d\over{d\xi}}\left(\xi^2\dpsi\right)\right]^{-1 } \nonumber \\   & = & { 1\over{r_0}}\left[1 - 9\xi\left(\dpsi\right)^{-1}{{\rho_s}\over{\rho_c } } \right]^{-1 } , \label{eq : a4}\end{aligned}\ ] ] where @xmath75 is the density just inside the cloud edge and we have used poisson s equation ( [ eq:2 ] ) . we also have , from equation ( [ eq : a2 ] ) , @xmath307 \nonumber \\   & = & p_s\left(\dpsi\right)^{-1}\left({{\rho_s}\over{\rho_c}}\right)\left [ 18-\left(\dpsi\right)^2\left({{p_s}\over{p_c}}\right)^{-1}\right ] , \label{eq : a5}\end{aligned}\ ] ] where both equations ( [ eq:1 ] ) and ( [ eq:2 ] ) have been used . thus , equations ( [ eq : a3 ] ) , ( [ eq : a4 ] ) , and ( [ eq : a5 ] ) show that @xmath308 . \label{eq : a5a}\ ] ] now , from equations ( [ eq:13 ] ) and ( [ eq:4d ] ) , we have @xmath309 so that finally , @xmath310 . \label{eq : a7}\ ] ] alternatively , for a _ nonmagnetic _ cloud , we can make use of equation ( [ eq:12]c ) , with @xmath300 , to write @xmath311 . \label{eq : a6}\ ] ] this treatment is valid for _ any _ gas equation of state , so long as the cloud radius is perturbed in such a way that @xmath72 and @xmath71 are unaffected . the result ( [ eq : a6 ] ) was also obtained by ebert ( 1955 ) and bonnor ( 1956 ) , who specifically considered the stability of truncated isothermal spheres , and by maloney ( 1988 ) in his study of negative - index polytropes .    any virialized cloud truncated at small @xmath187 ( i.e. , where @xmath133 and @xmath21 are approximately constant ) will be stable . this is because , regardless of the equation of state , the potential @xmath312 at small radii can always be expanded in a power series of the form : @xmath313 ( see appendix c for an example ) . thus , referring to equation ( [ eq : a5a ] ) with @xmath314 , we have @xmath315 . depending on the equation of state , there may or may not be a truncation at larger @xmath187 which results in @xmath316 and an unstable equilibrium . consider the ( nonmagnetic ) equation of state @xmath317 where the polytropic index @xmath131 . then we define @xmath134 and write @xmath318 with @xmath319 corresponding to isothermality . in order to evaluate the coefficients in equations ( [ eq:4a])([eq:4d ] ) ( or [ [ eq:12 ] ] , with @xmath300 ) , we first use the relation ( [ eq : b1 ] ) to integrate the equation of hydrostatic equilibrium ( [ eq:1 ] ) and find @xmath320^{n/(1-n ) } , \label{eq : b2}\ ] ] so that poisson s equation ( [ eq:2 ] ) becomes @xmath321^{n/(1-n)}. \label{eq : b3}\ ] ] ( note that in the limit @xmath322 , the right - hand side of eq .  [ [ eq : b3 ] ] becomes @xmath323 , just what is required for the isothermal sphere . ) at the cloud center , @xmath280 , so for small @xmath187 the potential @xmath312 can be expanded in a power series , @xmath324 allowing for an integration of equation ( [ eq : b3 ] ) . the result of this is a core - halo structure to the cloud : at very small radii , the solution is indistinguishable from that for an isothermal sphere , and the density and velocity dispersion are essentially constant with @xmath187 , independently of @xmath143 . once @xmath325 a few , however , a power - law density profile obtains ; for each @xmath143 , there exists a singular solution to which the equilibrium cloud tends at large radii . these singular solutions are readily found by substituting the prescription @xmath326 into equation ( [ eq : b2 ] ) and solving ( [ eq : b3 ] ) for @xmath327 and @xmath328 . this gives @xmath329^{p/2 } , \label{eq : b5}\ ] ] so that with @xmath330 we have @xmath331 , and these clouds are diffuse enough that they have no `` natural '' edge . that is , the density never vanishes , and any boundary to the cloud must be defined by the point at which the internal pressure matches that of an external medium . ( it is again worth noting the exact correspondence to the singular isothermal sphere when @xmath319 in eq . [ [ eq : b5 ] ] ; cf . chandrasekhar 1967 . ) consider now a polytrope of negative index , which is truncated at radius @xmath187 large enough that the structure of the cloud is given by the singular solution . then @xmath293 and @xmath242 can be calculated analytically , and equation ( [ eq:13 ] ) yields @xmath332 we also know that @xmath44 , so @xmath333 and we have the following : @xmath334 @xmath335 @xmath336 and @xmath337 thus , regardless of where they are truncated by the surface pressure @xmath97 , these model clouds can only ever represent modest enhancements over the intercloud medium . in particular , @xmath338 has a maximum of 3 ( for isothermality , @xmath319 ) , but even for @xmath157 ( which gives the @xmath133 dependence of alfvn wave pressure ; mckee & zweibel 1995 ) is reduced to just 9/7 . as discussed in  3 , any truncation of a negative - index polytrope results in a stable cloud . this is because , with @xmath141 , we have that @xmath339 , and the stability criterion ( [ eq : a7 ] ) becomes ( with the help of [ [ eq : b6 ] ] ) @xmath340 . in particular , this holds in the limit @xmath322 , when the numerator and denominator of ( [ eq : a7 ] ) both vanish . what allows for the existence of critical and unstable equilibria for the bounded isothermal sphere , then , is the fact that its density profile actually oscillates about the singular solution @xmath169 at large radii ( chandrasekhar 1967 ) . this does _ not _ occur for those polytropes with @xmath136 ; instead , the bounded spheres follow the singular solutions exactly at large @xmath187 .    finally , it is worth noting that these results do not contradict the well known instability ( e.g. , shu 1977 ) of truly singular spheres , for which @xmath141 all the way to the center and @xmath341 . in such cases , perturbation of the truncated cloud radius is performed with @xmath72 , @xmath71 , _ and _ @xmath181 all held fixed . stability then depends only on the sign of @xmath342 ( eq . 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we generalize the classic bonnor - ebert stability analysis of pressure - truncated , self - gravitating gas spheres , to include clouds with arbitrary equations of state . 
 a virial - theorem analysis is also used to incorporate mean magnetic fields into such structures . 
 the results are applied to giant molecular clouds ( gmcs ) , and to individual dense cores , with an eye to accounting for recent observations of the internal velocity - dispersion profiles of the cores in particular . 
 we argue that gmcs and massive cores are at or near their critical mass , and that in such a case the size - linewidth and mass - radius relations between them are only weakly dependent on their internal structures ; any gas equation of state leads to essentially the same relations . 
 we briefly consider the possibility that molecular clouds can be described by polytropic pressure - density relations ( of either positive or negative index ) , but show that these are inconsistent with the apparent gravitational virial equilibrium , @xmath0 , of gmcs and of massive cores . 
 this class of models would include clouds whose nonthermal support comes entirely from alfvn wave pressure . 
 the simplest model consistent with all the salient features of gmcs and cores is a `` pure logotrope , '' in which @xmath1 . 
 detailed comparisons with data are made to estimate the value of @xmath2 , and an excellent fit to the observed dependence of velocity dispersion on radius in cores is obtained with @xmath3 .