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it is not yet known what exactly induces and maintains galactic warps .
it may well be that a number of mechanisms  from among those proposed already , or some new ones 
act together or alone to produce this ubiquitous phenomenon .
some proposed mechanisms rely on a dark galactic halo as a direct actuator or as a mediator of perturbations ( for reviews with extensive references see @xcite , @xcite , and @xcite )
. the modified dynamics ( mond ) repudiates dark halos , but it offers a new mechanism that increases the warping efficacy of external perturbers over and above their possible tidal effects , which , notoriously , are too weak .
this results from the nonlinearity of mond and is most clearly demonstrated in the case where a system ( a galaxy ) falls in an external field that by itself is approximately constant in space . in a linear theory , such as newtonian gravity
, the constant external field has no effect on the internal dynamics of the system ( motions with respect to its center of mass ) ; in mond it very much does . when the external field dominates the internal field of the system it is easy to deduce what its effects are , as discussed e.g. in @xcite , and @xcite . in the present context
the external acceleration is small compared with the internal ones at the position of the warp , which necessitates numerical studies .
for our mechanism to work in field galaxies , one or more perturbers must be present .
there is , indeed , growing evidence that the appearance of a warp in a galaxy is strongly correlated with the presence of nearby perturbers ( see e.g. @xcite ) .
even galaxies that had been thought to be isolated might , in fact , not be so ( @xcite ) .
of course , perturber companions have always been suspected , but their direct tidal effects on disks seem to be too small .    the purpose of this letter is to demonstrate , by numerical solutions of simplified galaxy - perturber systems , that , with reasonable parameter values , this mond effect produces galactic warps of the magnitude observed .
we have not included effects due to variations in the external field ( due to the motion of the perturber , or to the motion of the galaxy in a parent cluster ) . from the symmetry of our model problem
the warps we produce have a straight line of nodes .
the method is described in section 2 ; the results are detailed in sections 3 and 4 ; conclusions are drawn in section 5 .
we use the nonrelativistic , modified - gravity formulation of mond suggested by @xcite . the acceleration field @xmath0 produced by a mass distribution @xmath1 is derived from a potential @xmath2 that satisfies @xmath3= 4\pi g\rho\ ] ] instead of the usual poisson equation @xmath4 , where @xmath5 for @xmath6 , and @xmath7 for @xmath8 , and @xmath9 is the acceleration constant of mond . the form
@xmath10 has been used in all rotation curve analyses , and we also use it here .
this nonlinear potential equation is solved numerically using multi - grid methods as detailed in @xcite , and adumbrated in @xcite .
we consider two classes of models . to simulate a far away companion , or the effect of the mean field of a cluster on a member galaxy , we solve for the field of a rigid disk in the presence of a given external acceleration field @xmath11 . in this case
the field equation is solved subject to the boundary condition at infinity @xmath12 .
then @xmath11 is subtracted from @xmath13 to get the field relative to the galaxy .
this latter determines the galaxy s internal dynamics , warps , etc . to simulate the effect of a nearby companion , exemplified here by the effect of the magellanic clouds ( mc ) on the milky - way ( mw )
, we solve fully for a disk - plus - perturber system ( in which case @xmath14 at infinity ) .
then , the center - of - mass acceleration of the galaxy is computed using the surface - integral method given by eq.(14 ) of @xcite , and is subtracted from the acceleration field to get the internal dynamics .    in each case , after the acceleration field relative to the center of mass of the galaxy is found , we find closed , nearly circular , nearly planar , test - particle orbits .
the orbits are integrated for many periods to insure that , within our accuracy and patience , they are closed .
thus , inasmuch as adiabaticity is a good approximation , these are non - precessing orbits .
they are also found to be stable under small changes in their initial conditions .
these are taken to trace a warp , in the spirit of the tilted - ring model ( @xcite ) .
we take the model galaxy to be an exponential disk smoothly truncated at a radius that we use as our unit length , @xmath15 , and with a scale length of @xmath16 in these units .
the surface density is of the form @xmath17 for @xmath18 .
the disk lies in the @xmath19 plane . to optimize the warping effect we take the external field to lie @xmath20 from the @xmath21 axis in the third quadrant of the @xmath22 plane .
its absolute value is taken as @xmath23 in units of @xmath9 .
( we work in units where @xmath24 , @xmath25 , so masses are given in units of @xmath26 . ) in a more extensive study we plan to calculate the effect as a function of the field direction ( relative to the disk axis ) and also to follow the test particle orbits as the external field changes with time to mimic the relative motion of the galaxy and perturber .
for the present , pilot study we ran models with two values of the disk mass @xmath27 and @xmath28 .
the ( mond ) accelerations of the isolated disk models at @xmath29 are @xmath30 ; i.e. respectively , ten and twenty times larger than the external field .
both accelerations are small compared with 1 ( @xmath9 ) so we are rather deep in the mond regime in the warp region . in the deep - mond regime
the theory has obvious scaling properties so the above parameters represent a family of models spanned e.g. by scaling by the same factor @xmath31 and @xmath32 , or @xmath33 and @xmath34 ( with @xmath35 fixed ) .
the results are summarized in figures [ fig1 ] and [ fig2 ] .
we first show for each model a plot of the absolute value of the torque @xmath36 in the @xmath22 plane containing the disk axis and the external field ( @xmath37 is the field in the center - of - mass frame ) .
this is a quantity that brings out clearly the departure of the field from both axisymmetry and left - right symmetry . in the spherical case @xmath38 everywhere ;
in the isolated - disk case the @xmath38 line is the @xmath21 axis ( and the @xmath39 axis ) .
this torque plot is also useful for homing in on closed orbits of the potential whose center is near the galactic center , because these should cross the @xmath22 plane near the zero - torque line .
the orbits are found by actual integration , starting from a set of initial conditions .
the projections of some such orbits are then shown .
we surmise that in the spirit of the tilted - ring model they delineate the shape of the warp .
the disk of the milky way is known to be warped beyond the solar circle ( @xcite , and for a recent description and references @xcite ) . at galactic longitude @xmath40 the hi disk curls steadily away from the plane . at @xmath41 the disk curves southward before turning back towards the plane ( see @xcite for an analytic expression that approximates the warp shape beyond 11 kpc ) .    the line of nodes in the tilted - ring picture is straight within the uncertainties , and is nearly perpendicular to the plane spanned by the inner - disk axis and the radius vector to the magellanic clouds .
this makes the cloud system a prime suspect in producing the warp .
it was , however , appreciated long ago that the tidal field of the clouds , in their present position , is too small to distort the disk to the extent observed .
for example , @xcite estimated that a cloud mass @xmath42 would generate a warp of amplitude @xmath43 70 pc at a radius of 16 kpc .
it has , however , been suggested by @xcite that a `` live '' halo that actively responds to the perturbation of the clouds might augment the effect to produce a warp of the observed magnitude and geometry .
mond , as we said , excludes a dynamically important halo , but might lead to a large enough warp due to the non - linear effect discussed above .
we model the system as follows .
the mw is taken as a pure disk in the @xmath19 plane , centered at the origin , with the cutoff , exponential surface - density law described in section 3 ( with @xmath16 ) ; its dimensionless mass is @xmath44 .
the magellanic clouds are represented by one point mass @xmath45 at a position whose dimensionless coordinates are ( 2.52 , 0 , -1.63 ) .
this is at @xmath46 from the center of the galaxy , and at the correct galactic latitude of the lmc .
this ratio would correspond for example to @xmath47 kpc ( @xcite ) and an lmc distance of 45 mpc ( @xcite ) .
the uncertainties in these parameters are still large .
two mass ratios were considered : @xmath48 .
( the b luminosity ratio of the clouds to the mw is about 0.2 .
since the baryonic @xmath49 values of the two might be different , reasonable values of the mass ratio lie between 0.1 and 0.4 . ) other nearby galaxies are expected to have a smaller effect than the lmc ; for example , m31 , despite its higher mass , causes a rather smaller acceleration near the mw .    in figure [ fig3 ] we show , for each mass ratio , two closed , quasi - circular , stable orbits beyond the cutoff radius of the disk  presumed to delineate our calculated warp  together with a representation of the observed warp ( as given by the formula in @xcite ) .
we see that for a constant external field whose ratio to the field of the isolated disk at @xmath50 is only ( 5 - 10)@xmath51 , a noticeable warp beyond this radius is indicated by test - particle orbits .
this acceleration ratio increases in proportion to the galactocentric radius , but , still , rotational velocities will be affected only little even up to @xmath52 . note that the warp is not symmetric but is less pronounced on the attracting side of the field .
we used an external - field inclination that is favorable for an s - shape warp .
when the field is in the disk plane , the axisymmetry is broken but the up - down one is not , so no warp will be induced .
if the field is perpendicular to the disk , axisymmetry is preserved but not the up - down one ; this might induce bowl - shaped warps .
( in the limit of a highly dominant , perpendicular , external field , the analytic results in @xcite show that the geometry remains up - down symmetric , but for a weak perturbing field this is not so . )    regarding the results for the mw - mc system , we see that even for a mass ratio of 0.1 a satellite at the position of the mc produces enough field distortion to accommodate inclined , quasi - circular orbit that rise to @xmath53 at @xmath54 on one side , and to a height of @xmath55 at @xmath56 on the other . with a mass ratio of 0.2 the amplitude of the warp is close to that observed for the mw .
. a larger volume of the parameter space has to be surveyed .
this includes more values of the relative strength of the perturbation , different disk - perturber alignments ( leading perhaps to a wider varieties of warp shapes ) , more complex perturbations such as two or more satellites , which would bend the line of nodes at larger radii , where we can not approximate the combined effect by a constant field .
we would also have to study other galaxy mass distributions .
for example , we expect that if a considerable fraction of the galaxy mass is put in a round bulge , a warp will form more easily . for the same reason , if @xmath57 is smaller ( but the mc distance remains the same ) the warp will be stronger at the same position .
. we must reckon with the fact that in many relevant cases the geometry of the perturbation changes considerably during the response time of the disk ( say the orbital period at the position of the warp ) .
this is true of galaxies moving in or near the core of galaxy clusters ; it is also true for the mw - mc system .
the warp geometry will thus not just follow the perturbation adiabatically but , at larger radii , the geometry may reflect the past history of the perturbation ( leading , among other things , to curvature of the line of nodes ) . according to proper - motion observations and models of the mc and the magellanic stream motion
( see e.g. @xcite ) , the mc binary is moving on a nearly polar orbit around the galaxy with a tangential velocity that is now comparable with the rotational velocity of the galaxy .
this means that the radius vector to the mc changes its angle with the galaxy s axis by @xmath58 during the galaxy s orbital period at about 15 kpc .
this means that the adiabaticity assumption we have made might be broken , and more and more so at larger radii .
our subtraction of a constant center - of - mass acceleration is then also not valid .
this could lead to a more complicated warp geometry than the integral - sign shape that we get with adiabaticity . because the mc orbit is nearly polar we expect the line of nodes to remain straight .
we think self - gravity of the mass in the warped region is not so important .
this is because the relative contribution of the warped mass to the acceleration field is small everywhere , even within the warp itself .
( the surface density in the warp is smaller than the integrated surface density there . )
this contribution can then be treated as a perturbation  linearizing in it the mond field equation  and in mond such density perturbations produce an even weaker effect than in newtonian dynamics ( hence the added stability in mond ) .
so we can at least expect that the nonlinearity of mond will not beget some peculiar amplification of self gravity .
but we do not really know what these effects might be  a point that has to be checked numerically .
we plan to perform @xmath59-body simulations whereby the @xmath60-body warped disk and the point - mass perturber orbit each other . this will account for non - adiabaticity and for self gravity in the disk , and also partly for point 2 above .
there is an effect in the modified dynamics ( mond ) that is conducive to formation of warps . because of the nonlinearity of the theory the internal dynamics of a galaxy 
 is affected by a perturber over and above possible tidal effects . 
 for example , a relatively distant and light companion or the mean influence of a parent cluster , with negligible tidal effects , could still produce a significant warp in the outer part of a galactic disk . 
 we present results of numerical calculations for simplified models that show , for instance , that a satellite with the ( baryonic ) mass and distance of the magellanic clouds can distort the axisymmetric field of the milky way enough to produce a warp of the magnitude ( and position ) observed . 
 details of the warp geometry remain to be explained : we use a static configuration that can produce only warps with a straight line of nodes . in more realistic simulations 
 one must reckon with the motion of the perturbing body , which sometimes occurs on time scales not much longer than the response time of the disk .