document: cataclysmic variables ( cvs ) are short - period binaries containing a white dwarf ( wd ) primary ( with mass @xmath2 ) and a low mass main sequence secondary ( with mass @xmath3 ) . 
   the secondary fills its roche lobe and transfers mass to the wd through the inner lagrangian ( @xmath4 ) point . 
   the main features of the orbital period distribution of cvs with hydrogen rich donors are the lack of systems in the 2 - 3 hr period range ( the so - called period gap ) and the sharp cut off of the distribution at around 77 minutes , as can be seen in figure  [ combined ] ( upper frame ; e.g. ritter & kolb 1998 ) . 
   so far theoretical models have been unable to reproduce the precise position of the observed short - period cut - off and observed shape of the cv orbital period distribution near this cut - off . 
   this is summarised in figure [ combined ] . 
   systems that evolve under the influence of gravitational radiation ( gr ; kraft et al . 
   1962 ) as the only sink of orbital angular momentum ( am ) reach a minimum period at @xmath5 minutes ( figure[combined ] , middle frame ; paczyski 1971 ; kolb & baraffe 1999).the probability of finding a system within a given period range is proportional to the time taken to evolve through this region . 
   we thus have n(p ) , for the number @xmath6 of systems found within a given orbital period range around @xmath7 , and @xmath8 is the secular period derivative at this period . 
   we thus expect an accumulation of systems ( a spike ) at @xmath9 where @xmath10 ( figure [ combined ] , lower frame ) , while no such spike is present in the observed distribution ( figure[combined ] , upper frame ) . 
   the orbital period evolution reflects the radius evolution of the mass donor , which in turn is governed by two competing effects . 
   mass transfer perturbs thermal equilibrium and expands the star . 
   thermal relaxation reestablishes thermal equilibrium and contracts the star back to its equilibrium radius . 
   the minimum period occurs where the two corresponding time scales , the mass transfer time @xmath11 and the thermal ( kelvin - helmholtz ) time @xmath12 are about equal ( e.g.  paczyski 1971 ; king 1988 ) . if @xmath13 then the star is able to contract in response to mass loss , but if @xmath14 the star will not shrink rapidly enough and will become oversized for its mass . 
   the position of the minimum period is therefore affected by the assumed mass transfer rate , and in particular by the assumed rate of orbital angular momentum ( am ) losses . in this paper 
   we investigate ways to increase the period minimum by increasing the mass transfer rate , and investigate ways to `` hide '' the spike by introducing a spread of @xmath9 values in the cv population . 
   in particular , we study the effect of a form of consequential am loss ( caml ) where the am is lost as a consequence of the mass transferred from the secondary , i.e. @xmath15 ( see e.g.  webbink 1985 ) . 
   in section [ theory ] we outline our general model assumptions and introduce the prescription for caml . in section [ sec22 ] 
   we present detailed calculations of the long - term evolution of cvs , and in section [ comptest ] we compare the observed short period cv period distribution with various theoretically synthesized model distributions based on the calculations in section 2 . 
   in this section we investigate possible solutions to the mismatch between the theoretical and observed minimum orbital period in cvs .      the orbital am loss rate @xmath16 of a cv 
   can be written as the sum of two terms ,    = _ sys+_caml ,    where @xmath17 denotes the `` systemic '' am loss rate , such as gravitational wave radiation , that is independent of mass transfer , while @xmath18 is an explicit function of the mass transfer rate . 
   we have    = 0    and    _ 
   caml0_20    we consider the general case in which the caml mechanism , along with nova mass ejections , causes a fraction of the transferred mass to leave the system . 
   this fraction may be greater than unity as the primary may lose more mass during a nova outburst than was accreted since the last outburst . 
   we employ a generic prescription of the effect of a caml mechanism , thus avoiding the need to specify its physical nature . 
   possible caml mechanisms include a magnetic propeller , i.e.  a system containing a rapidly spinning magnetic wd where some of the transferred material gains angular momentum from the wd spin by interaction with the wd s magnetic field ( see e.g.  wynn , king & horne 1997 ) , and an accretion disc wind ( see e.g. livio & pringle 1994 ) . 
   our caml prescription largely follows the notation of king & kolb ( 1995 ) . 
   the am is assumed to be lost via mass loss that is axis - symmetrical with respect to an axis a fixed at the wd centre but perpendicular to the orbital plane . 
   we define @xmath19 as the total fraction of mass lost from the secondary that leaves the system . 
   we assume further that a fraction @xmath20 ( @xmath21 ) of the transferred mass leaves the system with some fraction @xmath22 of the angular momentum it had on leaving the @xmath4 point . 
   we also consider mass that is lost from the system via nova mass ejections , which over the long term can be considered as an isotropic wind from the primary ( see e.g.  kolb et al . 
   this material will carry away the specific orbital angular momentum of the primary and will account for the fraction ( @xmath23 ) of the mass loss . 
   we thus obtain    _ caml = b^2_2 + ,    where we define @xmath24 as the caml efficiency . for comparison with king & kolb ( 1995 ) 
   we equate this to    [ eq : jdotcaml ] _ caml = j ,  >  0 , and obtain    [ eq : nufinal ] = ( 1+q)()^2 + . 
   for our calculations shown below we use the approximation    1-+-,^3=.    this is an adaptation of the expression given in kopal ( 1959 ) and is accurate to within 1% over the range @xmath25 .      in this subsection 
   we present calculations of the long - term evolution of cvs as they approach and evolve beyond the period minimum . for the computations we used the stellar code by mazzitelli ( 1989 ) , adapted to cvs by kolb & ritter ( 1992 ) . 
   some of these evolutionary sequences are the basis for the theoretical cv period distributions we present in section  [ comptest ] below . 
   we calculated the evolution of individual systems that are subject to caml according to equations [ eq : jdotcaml ] and [ eq : nufinal ] . 
   we chose @xmath26 and initial donor mass @xmath27 , with a range of caml efficiencies @xmath28 as shown in figure  [ fig : fullcaml ] . 
   the systems initially evolve from longer periods towards the period bounce ( right to left ) at almost constant mass transfer rate . the minimum period increases with increasing caml efficiency to a maximum of around 70 min for @xmath29 . 
   mass transfer stability sets an upper limit on the caml efficiency . 
   an obvious upper limit is 1 , where all the angular momentum of the transferred material is ejected from the system . although the ejected material may carry more angular momentum than was transferred ( as in the case of a propeller system where additional angular momentum is taken from the spin of the wd ) this 
   does not affect the net loss of orbital angular momentum . 
   the maximum caml efficiency still compatible with mass transfer stability could be smaller than unity . 
   the stability parameter @xmath30 which enters the expression for steady - state mass transfer , equation [ eq : stab ] ( e.g. king & kolb 1995 ) must be greater than zero ; this defines an upper limit on @xmath31 . 
   [ eq : stab ] -_2=m_2 ( )    a plot of @xmath30 against @xmath32 for an initially marginally stable system ( @xmath33 , @xmath34 and @xmath35 ) is given in figure [ fig : dq ] . 
   the system initially exhibits cycles of high mass transfer rate @xmath36 ( @xmath30 close to 0 ) and very low mass transfer rate @xmath37 . 
   the high states are short lived , on the order of @xmath38 years ( see figure [ fig : highmdot ] ) . 
   the system finally stabilizes with @xmath39 . at around @xmath40 @xmath30 
   starts to decrease further but always remains positive , settling at a value around @xmath41 . 
   the tidal deformation of the secondary may have an effect on the period minimum . 
   calculations by renvoiz , baraffe , kolb & ritter ( 2002 ) , [ see also kolb 2002 ] using 3dimensional sph models suggest that the secondary is deformed in the non - spherical roche lobe such that its volume  equivalent radius is around 1.06 times that of the same star in isolation . 
   we mimic this effect in our 1-dimensional stellar structure code by multiplying the calculated radius by a deformation factor @xmath42 before the mass transfer rate is determined from the difference between the radius and the roche lobe radius via    -_2=_0(- ) .    here 
   @xmath43 is the mass transfer rate of a binary in which the secondary just fills its roche potential and @xmath44 is the photospheric pressure scale height of the secondary ( see e.g.  ritter 1988 ) . 
   figure  [ fig : barraffe ] shows the effect on the minimum period and mass transfer rate for systems with various deformation factors @xmath42 , ranging from 1 ( no deformation ) to 1.24 . 
   the mass transfer rate is seen to decrease with increasing deformation . 
   this can be understood from the functional dependence on orbital period and donor mass in the usual quadrupole formula for the am loss rate due to gravitational radiation ( see e.g.  landau & lifschitz 1958 ) . 
   although the quadrupole formula is strictly valid only if both components are point masses , rezzolla , ury & yoshida ( 2001 ) found that the gr rate obtained using a full 3-dimensional representation of the donor star differs from the point  mass approximation by less than a few percent . 
   it can be seen from the figure that with the deformation factor 1.06 the minimum period increases from around 65 min to around 69 min , consistent with renvoiz et al ( 2002 ) for geometrical effects alone . a deformation factor of around 1.18 was required to raise the minimum period to the observed value of @xmath45 min . 
   this is somewhat larger than the intuitive expectation ( ) ^=()^=1.12 from kepler s law and roche geometry .    in our calculations 
   we consider the simple case in which only the geometrical deformation effects are taken into account . 
   the inclusion of the thermal effects considered by renvoiz et al ( 2002 ) have the likely effect of reducing @xmath9 , possibly by around 2% compared to the case with purely geometrical effects    one possible physical mechanism that could cause a deformation factor above the value of 1.06 is magnetic pressure inside the star , as suggested by dantona ( 2000 ) . 
   we note that patterson ( 2000 ) claims to find observational evidence for `` bloated '' secondaries in short period cvs . on the basis of donor mass estimates from the observed superhump excess period 
   he finds that the donors have @xmath46 larger radii than predicted from 1 dimensional . 
   , non  deformed stellar models if gravitational radiation is the only am sink . 
   even if true , this observation can not distinguish between an intrinsic deformation of the donor star or the non - equilibrium caused by orbital am losses in excess of the gr rate . 
   to test the statistical significance of the theoretically predicted accumulation of systems near the period minimum ( `` period spike '' ) we calculated the period distributions of model populations for various assumptions about evolutionary parameters . for each parameter a series of evolutionary tracks were generated , typically around 20 .    as systems evolve after the minimum period a point 
   is reached ( typically when @xmath47 falls below @xmath48 ) where numerical fluctuations in @xmath47 become so large that the henyey scheme no longer converges . 
   the stellar code uses tables to interpolate / extrapolate the opacities and equation of state for each iteration , and in this region the extrapolations become very uncertain . to extend the tracks we used a semi - analytical method as follows . 
   the tracks were terminated at a value of @xmath49 , where @xmath50 is the mass transfer rate at the minimum period for the track . 
   the radius of the star for the final part of the track is approximated by    r_2=r_0m_2^ ,    where @xmath51 and @xmath52 are assumed to be constant . 
   the values of @xmath51 and @xmath52 were determined from the final few data points for each track . 
   ( @xmath52 takes a typical value of around 0.15 for systems beyond the period bounce . ) to generate the extension to the track we then calculated @xmath7 from the roche lobe condition , and @xmath53 by assuming stationarity as in section [ minpnumeric ] ( see figure [ combined ] , middle frame for an example of an extended track ) . 
   we weight the chances of observation to the brighter systems by assuming    , 1.0 .    for the detection probability . 
   we tested the calculated model parent distributions for various values of the free parameter @xmath54 against the observed cv period distribution . a k - s 
   ( kolmogorov - smirnov ) test is insensitive to the differences between the parent distributions . 
   the greatest difference in the cumulative distribution functions ( cdfs ) of the observed and modelled distributions occur at the boundaries of the cdfs , i.e.  in the least sensitive region for the k - s test ( press et al 1992 ) . 
   we thus decided to use the following modified @xmath55 test . 
   for each parent distribution 10000 model samples each containing 134 systems were generated . ; 
   ritter & kolb 1998 , internal update june 2001 , as of july 2002 the number of systems in this period range is now 152 though this does alter the values given by the @xmath55 test , the trends and hence the results remain unaltered ] each sample was tested against the model parent distribution using a @xmath55 test , with 1 , 2 and 4 minute bins . 
   this range bridges the need for good resolution and significance of the @xmath55 test which requires a minimum number of cvs per bin . 
   the observed period distribution was tested against the model parent distribution also , giving the reduced @xmath55 value @xmath56 . 
   the fraction @xmath22 of generated samples with a reduced @xmath55 value less than @xmath56 was used as a measure of the significance level of rejecting the hypothesis that the observed distribution is drawn from the parent distribution . in the following we quote the rejection probability pr=@xmath22 . 
   kolb & baraffe ( 1999 ) noted that the observed distribution of non - magnetic cvs ( figure [ fig : mnmcvs ] , middle frame ) , and the observed distribution of magnetic cvs ( figure [ fig : mnmcvs ] , lower frame ) show no significant difference below the period gap . to test and quantify this we compared these distributions for @xmath57 min , giving a reduced @xmath55 probability of 0.1213 . 
   hence we can not rule out that the distributions are drawn from the same underlying parent distribution . 
   this is borne out by the results of comparing both distributions with a parent distribution that is flat in @xmath7 ( see also table [ tab : tabcomb ] , entries f and g ) which give similar rejection probabilities ( pr=0.709 and pr=0.781 , respectively ) . 
   we thus find no significant difference between the two distributions . in the following we therefore test models against the combined magnetic and non - magnetic distribution of observed systems . 
   the lack of any distinct features in the combined observed period distribution ( figure [ fig : mnmcvs ] , upper frame ) does indeed suggest an essentially flat distribution for the underlying parent distribution . 
   the flat distribution gives pr = 0.552 ( for the 1 minute bin width , see table [ tab : tabcomb ] ) . 
   we use this value as a benchmark for the models discussed below .    [ cols="^,^,^,^,^,^,^ " , ]      king , schenker & hameury ( 2002 ) constructed a ( nearly ) flat period distribution by superimposing individual idealized pdfs with different bounce periods @xmath58 according to a suitably tailored weighting . 
   for the double box - shaped idealised pdfs modelled on the pdf shown in our figure  [ combined ] ( lower frame ) the required weighting is @xmath59 $ ] ( @xmath60 is the observed minimum period ) . 
   this weighting function effectively mirrors the shape of the sharply peaked individual pdfs . 
   king et al .  ( 2002 ) found that the range @xmath61 is sufficient to wash out the period spike . 
   it is clear that this procedure involves a certain degree of fine - tuning for @xmath62 if the shape of the input pdf is given . 
   such a fine - tuning must surprise as the two functions involved presumably represent two very different physical effects . 
   we applied the weighting @xmath62 quoted in king et al . 
   ( 2002 ) to our non - idealized model pdfs that involve the caml efficiency and the deformation factor as a means to vary @xmath58 . 
   the weighting produced a marginally worse fit ( @xmath63 versus @xmath64 ; 1 minute binning ) for the caml pdfs compared to the parent population based on a flat caml efficiency spectrum we discussed earlier . in part this is due to the fact that the upper limit on @xmath31 does not allow a big enough range of @xmath58 . in the case of the deformation factor 
   pdfs the fit marginally improved ( @xmath65 versus @xmath66 ; 1 minute binning , @xmath67 ) . 
   it is possible to optimise the fit by adding systems with deformation factors up to 1.42 , and by using the weighting @xmath68 $ ] , but this still gives the fairly large value @xmath69 ( see also figure [ fig : kings ] ) . 
   however , such a parent population is inconsistent with the observed distribution for longer periods . 
   as can be seen from figure [ fig : barraffe ] systems that are subject to larger deformation factors would evolve into the period gap , hence the gap would be populated in this model .    for completeness 
   we show in figure [ fig : grsum1 ] the result of the superposition suggested by king , schenker & hameury ( 2002 ) if realistic rather than idealised pdfs are used . 
   this model assumes additional systemic am losses ( @xmath70 ; no caml , no deformation factor , @xmath71 ) as the control parameter for varying @xmath58 , and the weighting as in king et al . 
   the pronounced feature just above 2 hrs orbital period is the result of the adiabatic reaction of the donor stars at turn - on of mass transfer ( see e.g.  ritter & kolb 1992 ) . 
   such a feature is absent in the observed distribution . 
   if deformation effects are taken into account the additional am losses required to wash out the @xmath9 spike would cover a similar range but at a smaller magnitude . the resulting period distribution would be similar to the one shown in figure [ fig : grsum1 ] 
   we have investigated mechanisms that could increase the bounce period for cvs from the canonical theoretical value @xmath0 min to the observed value @xmath72 min , and ways to wash out the theoretically predicted accumulation of systems near the minimum period ( the period spike ) . 
   unlike king , schenker & hameury 2002 we focussed on effects other than increased systemic angular momentum ( am ) losses , i.e. we assume that gravitational radiation is the only systemic sink of orbital am . 
   we find that even a maximal efficient consequential am loss ( caml ) mechanism can not increase the bounce period sufficiently . as the real cv population is likely to comprise systems with a range of caml efficiencies we would in any case expect to have a distribution of systems down to @xmath0 min , rather than the observed sharp cut - off . 
   we considered donor stars that are `` bloated '' due to intrinsic effects , such as the tidal deformation found in 3-dim . 
   sph simulations of roche - lobe filling stars 
   . an implausibly large deformation factor of around 1.18 is needed to obtain a bounce period of @xmath45  min . 
   a possible alternative identification of @xmath9 as an age limit rather than a period bounce ( king & schenker 2002 ) would limit the donor mass in any cv in a cv population dominated by hydrogen 
   rich , unevolved systems to @xmath73 . 
   any system with donor mass much less than this would either have an orbital period less than 78 minutes or would have already evolved beyond the period minimum . 
   there are indeed systems with suspected @xmath74 ; good candidates are wz sge ( @xmath75 ; patterson et al 1998 ) and oy car ( @xmath76 ; pratt et al . 
   1999 ) . 
   it is also possible that systems die or fade before reaching the period bounce , and hence become undetectable as cvs . 
   the fact that the very different groups of non - magnetic and magnetic cvs show almost identical values of @xmath9 ( see figure [ fig : mnmcvs ] ) strongly suggests that the physical cause for the potential fading would have to be rooted in the donor stars or the evolution rather than the accretion physics or emission properties of the systems . 
   even if the bounce period problem is ignored we find in all synthesized model populations ( except for the age limit model ) a pronounced remaining feature due to the accumulation of systems near the bounce . we employ a modified @xmath55 test to measure the `` goodness '' of fit against the observed sample . an f - test ( press et al 1992 ) was also applied to the majority of @xmath77 models and the same general trends observed . 
   none of our synthesised model populations fits as well as the distribution which is simply flat in orbital period ( rejection probability @xmath78 ) . 
   only models where brighter systems carry a far greater weight than expected in a simple magnitude  limited sample ( selection factor @xmath79 with @xmath80 rather than @xmath81 ) achieve similar values for @xmath82 . 
   however , most of our models with @xmath83 canot be rejected unambiguously on the basis of this test . 
   models designed to `` wash out '' the period spike by introducing a large spread of the caml efficiency do generally better than population models based on donor stars that are subject to a large spread of intrinsic deformation factors . for all models 
   the rejection probability decreases if the full wd mass spectrum is taken into account , as this introduces an additional spread in the bounce period . 
   model populations where all cvs form at long orbital periods ( chiefly above the period gap ) give a much better fit than models that include newborn cvs with small donor mass . adding these systems to the population introduces a general increase of the orbital period distribution towards short periods , thus making the period spike more pronounced . 
   this suggests that most cvs must have formed at long periods and evolved through the period gap to become short - period cvs . 
   this is consistent with independent evidence that cv secondary stars are somewhat evolved ( baraffe & kolb 2000 ; schenker et al . 
   2002 ; thorstensen et al 2002 ) . 
   recently , king , schenker & hameury ( 2002 ) constructed a flat orbital period distribution by superimposing idealised pdfs that describe subpopulations of cvs with a fixed initial donor mass and initial wd mass , but different bounce periods . 
   this superposition required a strongly declining number of systems with increasing bounce periods . 
   we repeated this experiment with a realistic pdf , but failed to obtain a markedly improved fit .    in conclusion , we find that the period minimum problem and the period spike problem remain an open issue . 
   it is possible to construct cv model populations where the period spike is washed out sufficiently so that it can not be ruled out unambiguously on the basis of an objective statisticial test against the observed cv period distribution . 
   we thank graham wynn , andrew king and isabelle baraffe for useful discussions . andrew conway and chris jones who gave advice on the statistical analysis . 
   we also thank andrew norton for a critical reading of the paper and the referee jean - marie hameury for useful comments .
summary: we investigate if consequential angular momentum losses ( caml ) or an intrinsic deformation of the donor star in cvs could increase the cv bounce period from the canonical theoretical value @xmath0 min to the observed value @xmath1 min , and if a variation of these effects in a cv population could wash out the theoretically predicted accumulation of systems near the minimum period ( the period spike ) . 
   we are able to construct suitably mixed cv model populations that a statisticial test can not rule out as the parent population of the observed cv sample . 
   however , the goodness of fit is never convincing , and always slightly worse than for a simple , flat period distribution . generally , the goodness of fit is much improved if all cvs are assumed to form at long orbital periods . 
   the weighting suggested by king , schenker & hameury ( 2002 ) does not constitute an improvment if a realistically shaped input period distribution is used . 
   binaries : close  stars : evolution  stars : mass - loss 
   novae , cataclysmic variables .