You are an expert at summarizing long articles. Proceed to summarize the following text: 
one of eddington s famous discoveries is that the isotropic distribution function ( df ) of a spherical stellar system can be calculated from the density using an abel transform pair @xcite : @xmath6 here , @xmath7 is the binding energy per unit mass , and @xmath8 is the relative potential .    however , galaxy halos produced in cosmological simulations are held up by anisotropic velocity dispersions ( see e.g. , * ? ? ? the recovery of an anisotropic df for a spherical system is a more difficult problem @xcite . now , the steady state df can depend not only on @xmath9 but also on the magnitude of the specific angular momentum @xmath10 through jeans theorem . here @xmath11 is the two - dimensional velocity component projected on the tangential plane .    at least in the comparison of observational data with models ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , a widely used _ ansatz _ for the form of df is @xmath12 where @xmath13 is a function of the binding energy alone . it is easy to show that the models generated by this df exhibit the property that @xmath14 here , @xmath15 and @xmath16 are the radial and the tangential velocity second moments . the general inversion formula for the unknown function @xmath13 ( * ? * ; * ? ? ? * ; * ? ? ? * see also appendix  [ sec : con ] ) is no more difficult than eddington s formula ( eq .  [ eq : eddington ] ) . we note that it is common practice to characterize the velocity anisotropy of a spherically symmetric stellar system by the parameter @xmath17 defined in equation ( [ eq : beta ] ) , which is sometimes referred to as the anisotropy parameter @xcite . if @xmath18 , the velocity dispersion ellipsoid is a prolate spheroid with the major axis along the radial direction ( radially anisotropic ) , while it is an oblate spheroid with the tangential plane being the plane of symmetry ( tangentially anisotropic ) if @xmath19 . in particular , @xmath20 and @xmath21 indicate that every star is in a radial or circular orbit , respectively . on the other hand , systems with isotropic velocity dispersions have @xmath22 . in general , the anisotropy parameter varies radially , but for the system with a df of the form of equation ( [ eq : ansatz ] ) , it is constant everywhere . despite the attractiveness of a simple inversion of the df in equation ( [ eq : ansatz ] ) , the assumption of uniform anisotropy is not always desirable . for example , it is useful to be able to build models that are isotropic in the center and radially anisotropic in the outer parts , as seems to be the case for dark matter halos @xcite . dfs that are tangentially anisotropic in the outer parts are helpful for understanding the substructure and the satellite galaxy populations in dark matter halos . in this paper we study algorithms for building galaxies with varying anisotropy , using sums of dfs of the form of equation([eq : ansatz ] ) . [ sec : gis ] introduces a new family of cusped halo models , the generalized isochrones , and   [ sec : gp ] examines the generalized plummer models . every member of the family has analytic potential - density pairs , velocity dispersions , and dfs . specific examples are provided with the cosmologically important @xmath0 , @xmath2 , and @xmath1 density cusps . [ sec : app ] discusses a specific application of our theoretical models to the problem of estimating the mass of a host galaxy from the motion of its satellites . finally ,   [ sec : conc ] concludes with a brief comparison between our methods of producing varying - anisotropy dfs and other suggestions in the literature ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the isochrone sphere has the potential - density pair @xcite @xmath23 where @xmath24 is the total mass of the system , @xmath25 , and @xmath26 is a constant defining the scale length . like the @xcite model , the density is finite and well behaved at the center . unlike the plummer model , the density falls off as @xmath27 at large radii , which is a better fit to elliptical galaxies than the @xmath4 fall - off of the plummer model . there are many available studies regarding its dfs ( e.g. , * ? ? ? * ; * ? ? ? * ) , some of which are essentially numerical in nature .    inspired by this , let us consider the potential - density pair @xmath28 where @xmath29 here , @xmath30 is a new parameter used for the generalization of the isochrone sphere ( eq .  [ eq : iso ] ) . the condition that the density decreases monotonically with increasing radius restricts @xmath31 , with the limit ( @xmath32 ) being the cored classical isochrone sphere . in fact , all the halo models with @xmath33 are cusped . in particular , the density behaves as @xmath34 as @xmath35 . on the other hand , we find that @xmath36 where @xmath37 as @xmath38 . the density profiles of some members of this family are shown in figure  [ fig : geniso ] . we note that the case @xmath39 , for which @xmath40 , is actually recognized as the @xcite model , albeit with a slightly unusual identification for the scale length ( @xmath41 ) . models with @xmath42 and @xmath43 possess the cosmological cusps suggested by @xcite and @xcite .    the enclosed mass and the circular speed are found to be @xmath44^{-2 } \,;\qquad v_\mathrm c^2=\frac{gm_r}r=\frac{gmr^p}{s^{p-1}(b+s)^2}.\ ] ] the potential @xmath45 of the generalized isochrone sphere is simple enough that the inverse , @xmath46 , can be easily found ( with @xmath47 without any loss of generality ) : @xmath48^{1/p}}\psi.\ ] ] using this , in principle , one can construct infinitely many different expressions for @xmath49 , each of which can be inverted to yield the df by means of integral transforms @xcite . however , for most cases this procedure is analytically intractable . rather , it is often more productive to look for a specific form of @xmath49 that is associated with a plausible _ ansatz _ for the df , whose inversion reduces to elementary integrals . one such possible ( @xmath50,@xmath8)-split with an analytically tractable inversion is given by @xmath51 we note that a df of the form of equation ( [ eq : ansatz ] ) can be inverted from an ( @xmath50,@xmath8)-split that is a monomial of @xmath50 , with its power index linearly related to @xmath17 . by having the ( @xmath50,@xmath8)-split given by a linear combination of two monomials of @xmath50 , the inversion of equation ( [ eq : expone ] ) can be achieved by simply extending the known procedure , while the system is no longer restricted to have uniform velocity anisotropy . inspired by this , let us suppose that the df is of the form @xmath52 where @xmath53 and @xmath54 are functions of @xmath9 to be determined . then , the density is found to be @xmath55 where @xmath56 is the gamma function . by comparing this to equation ( [ eq : expone ] ) , a possible df may be found by inverting the integral equations @xmath57 the inversion procedure is essentially identical to the one that leads to equations  ( [ eq : disint ] ) and ( [ eq : dishalf ] ) . after this inversion , we find that @xmath53 and @xmath54 are @xmath58 both of which are always nonnegative for all accessible @xmath9-values ( @xmath59 ) . while we have given the result in complete generality in terms of hypergeometric functions @xmath60 , the dfs reduce to entirely elementary and analytic functions if @xmath42 , 1 , or 2 . in fact , if @xmath42 , both hypergeometric functions are just unity , whereas if @xmath61 is an integer , both reduce to elementary functions . if @xmath61 is a half integer other than @xmath62 , only @xmath63 reduces to an elementary function .    of particular interest is the case in which @xmath42 , for which the model possesses a central cusp like @xmath64 , as found by cosmological simulations @xcite . the corresponding df is very simple , namely , @xmath65 if @xmath39 ( the hernquist model with a rescaled scale length ) , @xmath66 + \frac1{4\pi^3l}\frac{e^2(3 - 2e)}{(1-e)^2},\ ] ] or if @xmath32 ( the classical isochrone sphere ) ; @xmath67 \nonumber\\&+ & \frac3{2^{15/2}\pi^3(1-e)^4 } \left[(16e^2 - 12e+3)\,\frac{15\arcsin\!\sqrt e}{\sqrt{1-e } } -\left(45 - 150e+144e^2 - 208e^3 + 64e^4\right)\sqrt e\right].\end{aligned}\ ] ] both of these dfs are no more complicated than the isotropic dfs deduced by @xcite and @xcite , respectively .      to find the velocity dispersion of these models , we exploit the splitting @xmath68 , where @xmath69 then , @xmath70 where @xmath71 the anisotropy parameter @xmath72 \label{eq : gicvi}\ ] ] is now no longer constant . figure  [ fig : anisoc ] shows the behavior of @xmath17 for some of these models . the curves can be understood on examination of equation ( [ eq : gicvi ] ) . this reveals that whereas @xmath73 at @xmath74 for any @xmath61 , @xmath75 if @xmath76 , or @xmath77 if @xmath78 . at the critical value @xmath39 , we find that the anisotropy parameter monotonically decreases from @xmath79 at @xmath74 to @xmath80 as @xmath38 . hence , the dfs discussed in this section build the generalized isochrone spheres with either decreasing @xmath17 in the inner region and increasing @xmath17 in the outer region ( @xmath78 ) , or decreasing @xmath17 throughout ( @xmath81 ) . the particular case of the latter also includes the hernquist model ( @xmath39 ) and the classical isochrone sphere ( @xmath32 ) . the simplest case is again @xmath42 , which leads to @xmath82 then we find that the virial theorem holds locally for this case : @xmath83 and the anisotropy parameter @xmath84 varies from @xmath85 at @xmath74 , to @xmath86 at @xmath87 , and back to @xmath88 as @xmath38 . this model in fact corresponds to one of the generalized hypervirial models studied by @xcite . that is , @xmath89^{-1}=(2 + 2r^{1/2}+r)^{-1}$ ] so that @xmath90 and @xmath91 in their parameters . we note that both @xmath92 and @xmath93 in fact reduce to the incomplete beta functions , which may be useful for the purpose of numerical calculations . furthermore , if @xmath94 is an integer , both @xmath92 and @xmath93 reduce to elementary functions of @xmath8 , and so do the velocity second moments . in particular , if @xmath39 [ for which @xmath95 , @xmath96 @xmath97 next , let us consider the one - parameter family of potential - density pairs , @xmath98 which was originally introduced by @xcite . this includes the @xcite model ( @xmath39 ) and the @xcite model ( @xmath32 ) as particular cases . @xcite recently found that this family can be constructed from power - law dfs of the form of @xmath99 . here we construct slightly more complicated dfs that build this family of models . we first find every df of the form of equation ( [ eq : ansatz ] ) , and then use them to build models with a more general variation of anisotropy .      with @xmath100 , we find that @xmath101 and that @xmath102 for @xmath103 . then , equation ( [ eq : disint ] ) reduces to @xmath104 where @xmath105 is the integer floor of @xmath106 , and @xmath107 . if @xmath108 , it is at least formally possible to derive the series expression of the df for arbitrary @xmath61 and @xmath17 from the integral form , namely , @xmath109 which becomes the generalized hypergeometric series if @xmath61 is a rational number . on the other hand , if @xmath110 , we find that @xmath111 so that the corresponding df is unphysical . hence , for the potential - density pair of equation ( [ eq : pdpair ] ) , the constant anisotropy df is physical only if @xmath112 . this means that the hypervirial models of @xcite , which have an anisotropy parameter @xmath73 , have the maximally radially biased velocity dispersions for a given @xmath61 and a constant @xmath17 . although there exist models with @xmath17 locally exceeding @xmath113 for the generalized plummer sphere of given @xmath61 ( see e.g. , * ? ? ? * ) , @xmath17 at the center can not be greater than @xmath113 for physically valid dfs . this is in agreement with the cusp slope central anisotropy theorem derived by @xcite . the corresponding velocity dispersions can be found either from equation ( [ eq : vint ] ) or by solving jeans equation . the results can be generally expressed using the incomplete beta functions . the equivalent results expressed in terms of hypergeometric functions are also found in equation ( 9 ) of @xcite .    for particular values of @xmath61 or @xmath17 , the df of equation ( [ eq : gpdf ] ) ( or equivalently , eq .  [ eq : gpdfs ] ) reduces to a simpler form . for example , if @xmath79 , the df is ( see eq . [ eq : dishalf ] ) @xmath114,\ ] ] where the positive definiteness of the df implies that @xmath115 . similarly , for @xmath116 , the df is @xmath117,\ ] ] which is nonnegative for @xmath118 if @xmath119 . on the other hand , the df of the hernquist model with constant @xmath17 reduces to the hypergeometric function ( see e.g. , * ? ? ? * ) @xmath120 which is nonnegative everywhere if @xmath121 , whereas that for the constant-@xmath17 plummer model reduces to @xmath122 which is physical if @xmath123 . since equation ( [ eq : gpd ] ) is valid for any constant @xmath17 with common @xmath124 , it is , in general , also possible to express the density profile of the generalized plummer model by @xmath125 where the parameters @xmath126 are normalized weights . consequently , the superposition of the constant anisotropy dfs , @xmath127 , still builds the generalized plummer sphere . here each @xmath128 is given by equation ( [ eq : gpdf ] ) ( or equivalently , eq .  [ eq : gpdfs ] ) in which @xmath17 is replaced by @xmath129 . the velocity second moments for these models are found to be @xmath130 where @xmath131 and therefore the anisotropy parameter @xmath17 is no longer constant .    for the particular case in which the sum only contains two terms ( such that @xmath132 , @xmath133 , and @xmath134 for all other @xmath135 values ) , the anisotropy parameter is @xmath136 which is a decreasing function of @xmath50 , provided that @xmath137 and @xmath138 . if we assume @xmath139 and @xmath137 , the limiting values are found to be @xmath140 on the other hand , @xmath141 or @xmath142 the simplest example of the dfs of this kind is obtained when we choose @xmath143 and @xmath144 : @xmath145 , \label{eq : sdf } \ ] ] which is physical if @xmath146 . the behavior of the anisotropy parameter @xmath17 for the model given by dfs of equation ( [ eq : sdf ] ) is shown in figure  [ fig : aniso ] . again we note that equation ( [ eq : hypg ] ) in general reduces to the incomplete beta function while , for this case , it further reduces to an expression involving only elementary functions of @xmath8 if @xmath147 is an integer , and so and so do @xmath17 and the velocity second moments . as an application of the preceding theory , let us consider the problem of estimating the mass of the milky way from the radial velocities of its satellites . there are 27 distant globular clusters and dwarf galaxies at galactocentric distance @xmath50 greater than 20 kpc ( see e.g. , tables 2 and 3 of * ? ? ? these range from arp  2 at @xmath148 out to leo  i at @xmath149 . the median distance of the satellite population is @xmath150 . the number density of the satellites falls off roughly like @xmath151 . this dataset is often used to estimate the mass of the milky way @xcite , as the h1 rotation curve can not be traced beyond @xmath520 kpc . the satellites are therefore the only available probe of the gravitational field of the distant halo . all the satellites have accurate radial velocities , but proper motions are available only for a handful of the closest objects . of course , the heliocentric radial velocity and the galactocentric radial velocity are basically the same ( once the motion of the sun has been taken out ) , as the galactocentric distance of the sun is much smaller than those of the satellites . the kinematic data therefore constrains the mean kinetic energy in the radial direction @xmath152 . again using the data in tables 2 and 3 of @xcite , we find that @xmath153 .    to model the milky way halo , we assume that its potential and density take the form of either the generalized isochrone or the generalized plummer models . furthermore , we assume that the number density of satellites shadows the total density ( e.g. , * ? ? ? * ; * ? ? ? this means that the satellites are drawn from dfs of the form given by either equation ( [ eq : diffiso ] ) or equation ( [ eq : sdf ] ) . we set the scale length so that the half - mass radius of the assumed density profile corresponds to the observed median radius of the population . the mass of the milky way @xmath24 enters into the potential and the df . it can therefore be constrained by requiring that the mean kinetic energy in the radial direction be consistent with the data , viz , @xmath154 the parameter @xmath61 in the dfs controls the variation of the velocity anisotropy with radius . hence , equation ( [ eq : crux ] ) defines a line in the plane @xmath155 , which is the _ mass - anisotropy degeneracy curve _ for the model . figure  [ fig : massaniso ] , inferred from numerical integrations , shows such curves computed for a number of the generalized isochrones and the generalized plummer models .    in this problem , the mean kinetic energy in the radial direction is fixed . models with tangential anisotropy at large radii therefore lead to a higher inferred mass for the milky way galaxy . the relative location of the curves in figure  [ fig : massaniso ] can be understood by examining the behavior of the anisotropy parameter for the models . more importantly , as the velocity anisotropy changes , the inferred mass of the milky way ranges from @xmath156 to @xmath157 . now , the velocity anisotropy of the satellite galaxy population is unknown , so the mass - anisotropy degeneracy _ by itself _ enforces a factor of @xmath53 uncertainty in the mass of the milky way . this fundamental limitation can only be overcome by proper - motion data . of course , there are other uncertainties ; for example , the half - mass radius of the satellite population used to set the scale length in these calculations is also not well known ( note that the locations of the satellites are likely to be biased toward their apogalacticon ) and has an uncertainty of @xmath520% . this carries over into an additional @xmath520% uncertainty in the inferred mass . this is small in comparison with the mass - anisotropy degeneracy , but not insignificantly so . this paper provides a number of flexible and simple galaxy models extended from two classical cored profiles : the isochrone sphere and plummer model . we also derive analytic distribution functions ( dfs ) with varying velocity anisotropy for all the models including new dfs for the classical isochrone sphere and the hernquist model . dfs of the form @xmath158 build a spherical system with a constant anisotropy parameter @xmath17 , where @xmath9 is the binding energy and @xmath159 is the specific angular momentum . the assumption of constant anisotropy is not enough to provide realistic models , as both observational data and numerical simulations suggest that dark halos and their constituents ( such as satellites ) have velocity anisotropies that vary with radius ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? here we have explored models whose dfs are superpositions of two or more such terms , namely , @xmath160 this can be done in two ways . first , as illustrated by the generalized isochrones in   [ sec : gis ] , we can sometimes find a splitting of the density in the form @xmath161 and apply the inversion separately to each component . for the generalized isochrones , this provides models with either decreasing anisotropy parameter @xmath17 in the inner region and increasing @xmath17 in the outer parts , or decreasing @xmath17 throughout . second , as illustrated by the generalized plummer models in [ sec : gp ] , we can always add together weighted sums of constant anisotropy dfs , each of which individually reproduces the required density . this appears to provide models with decreasing anisotropy parameter @xmath17 throughout . at large radii , the df is dominated by the component of smallest @xmath17 , and the models can be either radially or tangentially anisotropic in the outer parts , according to choice . for both cases , it appears that the anisotropy parameter always decreases on beginning to move outward from the very center . there have been a number of previous algorithms suggested for building anisotropic dfs with varying anisotropy . for example , the osipkov - merritt algorithm @xcite builds models that are isotropic in the inner parts and tend to extreme radial anisotropy ( @xmath162 ) in the outer parts . @xcite provided a clever algorithm for building separable dfs of the form @xmath163 where @xmath164 is the angular momentum of a circular orbit with energy @xmath9 . the analytic examples provided by @xcite all have increasing radial anisotropy in the outer parts . his method can be used to generate tangentially anisotropic models , but usually at the cost of a numerical inversion . the methods developed in this paper therefore provide a complement to the existing inversions . tangential anisotropy often arises in accreted populations , such as satellite galaxies . as an application of our models , we have shown that the mass - anisotropy degeneracy by itself provides a factor @xmath53 uncertainty in the mass of the milky way galaxy as deduced from the kinematics of its satellites . the density profiles of models studied in this paper typically fall off faster than @xmath165 and slower than @xmath166 at large radii , which are in good agreement with both observations of spheroidal components of galaxies and results from numerical simulations . at small radii , they are typically cusped , with cusp indices in the range suggested by numerical simulations . in particular , there are members with density cusps like @xmath167 , @xmath168 , and @xmath169 , which have been suggested as important on cosmogonic grounds @xcite , although observational evidence seems to favor the presence of a constant - density core @xcite . however , even if real galaxies are cored , the core would be such a small fraction of the halo that it is not practically important for studies of the whole . our families of new models therefore should find widespread application in the modeling of galaxies and dark halos . we thank c. hunter , who made a number of interesting comments on the various incarnations of this paper . an , j. , & evans , n. w. 2005a , , 444 , 45 ( astro - ph/0508419 ) an , j. h. , & evans , n. w. 2005b , , submitted ( astro - ph/0511686 ) baes , m. , & dejonghe , h. 2002 , , 393 , 485 bertin , g. , leeuwin , f. , pegoraro , f. , & rubini , f. 1997 , , 321 , 703 binney , j. , & tremaine , s. 1987 , galactic dynamics ( princeton nj : princeton univ . press )    cuddeford , p. 1991 , , 253 , 414 de blok , w. j. g. , mcgaugh , s. s. , bosma , a. , & rubin , v. c. 2001 , , 552 , l23 dejonghe , h. 1987 , , 224 , 13 donato , f. , gentile , g. , & salucci , p.  2004 , , 353 , l17 eddington , a. s. 1916 , , 76 , 572 eggen , o. j. , lynden - 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anisotropy dfs used in the main body of the paper . for the df given in equation ( [ eq : ansatz ] ) , direct integration over velocity space gives @xmath170 the formula for inversion for the unknown function @xmath13 is given by @xcite @xmath171 , \label{eq : disint}\end{aligned}\ ] ] where @xmath172 is expressed as a function of @xmath8 , and @xmath105 and @xmath173 are the integer floor and the fractional part of @xmath106 , respectively . this includes equation ( [ eq : eddington ] ) as a particular case of @xmath22 ( @xmath174 & @xmath175 ) . if @xmath17 is a half - integer constant ( i.e. , @xmath176 @xmath62 , @xmath177 , @xmath178 , and so on ) , the above inversion simplifies to @xmath179 that is , this only involves differentiations in the process @xcite . in addition , the expression for the differential energy distribution ( ded ) is found to be @xmath180 @xmath181 where @xmath182 , and @xmath183 and @xmath184 are the dirac delta ` function ' and the heaviside unit step function , respectively . in particular , if @xmath79 , the last integral does not explicitly involve the potential , and so the ded may be found simply as @xmath185 . it is also straightforward to find the integral for the second velocity moments ; @xmath186 in fact , the velocity dispersion of any constant anisotropy models may be found from @xmath187 without explicitly evaluating the df @xcite . on differentiating equation ( [ eq : diffy ] ) with respect to @xmath8 and noting that the right - hand side is the same as equation ( [ eq : den ] ) , that is , @xmath188 , it follows that @xmath189 we note that equation ( [ eq : vint ] ) can also be derived from the integral solution of jeans equation with a constant @xmath17 ( see eq . 9 of * ? ? ? this also indicates that equation ( [ eq : vint ] ) is actually a general result for a constant @xmath17 model , regardless of the specific form of the df that generates the model .    as an example , let us think of the dfs that build the generalized isochrone sphere with a constant anisotropy parameter . first , the df and the ded for the generalized isochrone sphere with @xmath79 can be found from equation  ( [ eq : dishalf ] ) ; @xmath190^{1/p } } \,;\qquad \frac{dm}{de}= \frac{\left[(1-e)^p - e^p\right]^{1/p}}{2(1-e)^{2p}}\ g(e)\ ] ] @xmath191 + ( 1-e)^pe^{p-1}\left[(p+1)(p+2)+12e^2 - 6(p+2)e\right ] + 6(1-e)^{2p+1}. \ ] ] here , because @xmath59 , the nonnegativity of the df is satisfied only if @xmath115 . in addition , the integration involved in the inversion ( eq .  [ eq : disint ] ) is also analytically tractable if @xmath73 . that is , for the density profile of the generalized isochrone sphere , we find that @xmath192.\ ] ] this leads to the df of the form of equation ( [ eq : ansatz ] ) @xmath193,\qquad\end{aligned}\ ] ] which build a constant anisotropy model with @xmath73 . if @xmath39 ( corresponding to the hernquist model ) and @xmath79 , the df and the ded further reduce to a particularly simple form ; @xmath194 which was originally found by @xcite for the @xcite model . for @xmath32 and @xmath22 , the well - known expression for the df of the isotropic isochrone sphere ( see e.g. , * ? ? ? * ; * ? ? ? * ) is recovered ; @xmath195.\ ] ] finally , let us think of the ded of the generalized plummer models corresponding to the df given in equation  ( [ eq : gpdf ] ) . for this case , equation  ( [ eq : ded ] ) reduces to a similar integral as equation  ( [ eq : gpdf ] ) @xmath196 like the df , equation  ( [ eq : dedgp ] ) reduces to a closed form function of @xmath9 for particular values of @xmath61 or @xmath17 . among them are for @xmath79 @xmath197e^{p-1 } , \label{eq : dedpp}\ ] ] for @xmath116 , @xmath198e^{p-1 } { \ensuremath{\mbox{}_2f_1\!\left(1,\frac{1}{p}+1;\frac{4}{p}+2;1-e^p\right)}},\qquad \label{eq : dedpn}\end{aligned}\ ] ] and for @xmath39 , @xmath199 this suggests that @xmath200 diverges as @xmath201 for @xmath78 , that it vanishes for @xmath76 , and that it has a finite limiting value of @xmath202 for @xmath39 . we believe that this is because the behavior of @xmath200 near @xmath203 is dominated by the stars at large radii and the @xmath27 fall - off marks a critical point @xcite .
this paper provides two families of flexible and simple galaxy models . 
 many representatives of these families possess important cosmological cusps , with the density behaving like @xmath0 , @xmath1 , or @xmath2 at small radii . 
 the density falls off between @xmath3 and @xmath4 at large radii . 
 we provide analytic and anisotropic distribution functions for all the models . unlike many existing methods , 
 our algorithm can yield tangentially anisotropic velocity dispersions in the outer parts , and so is useful for modeling populations of satellite galaxies and substructure in host galaxy halos . as an application , we demonstrate the degeneracy between mass and anisotropy for the satellite galaxy population of the milky way . 
 this can introduce a factor of @xmath53 uncertainty in the mass of the milky way as inferred from the kinematics of the satellite population .