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microlensing is one of the most powerful methods that can be used to search for extrasolar planets @xcite . recently , two robust microlensing detections of exoplanets were reported by @xcite and @xcite . the signal of a planetary companion to microlens stars is a short - duration perturbation to the smooth standard light curve of the primary - induced lensing event occurring on a background source star . the planetary perturbation occurs when the source star passes close to the caustic . the caustic represents the set of source positions at which the magnification of a point source becomes infinite . studies of the properties of the caustic are important because the characteristics of the planetary perturbations in the microlensing light curve depend critically on the properties of the caustic . for example , the location of the perturbation on the lensing light curve depends on the location of the caustic . the duration of the perturbation and the probability of detecting the perturbation are proportional to the caustic size . in addition , the pattern of the perturbation is closely related to the caustic shape . therefore , it is essential to understand the properties of caustics for the interpretation of the planetary lensing signals . although some of the properties of the caustics in planetary microlensing have been known , our knowledges of them are mostly from scattered information based on numerical approaches . the problem of the numerical approach is that the dependence of the planetary lensing behavior on the planet parameters of the star - planet separation @xmath0 ( normalized by the einstein ring radius @xmath1 ) and the planet / star mass ratio @xmath2 is not clear . there have been several attempts to overcome this ambiguity using analytic methods . by treating the planet - induced deviation as a perturbation , @xcite and @xcite derived analytic expressions for the locations of the _ central _ caustic , which is one of the two sets of caustics of the star - planet lens system located close to the primary star . based on a similar perturbative approach , @xcite provides analytic expressions for the locations of the lensing images . @xcite derived analytic expressions for the locations of not only the central caustic but also the _ planetary _ caustic , the other set of caustics , which are located away from the central star . however , there has been no analytic work on the detailed properties of the caustics such as the location , size , and shape , except the very recent work of @xcite ( hereafter paper i ) on the central caustics . following paper i , we conduct a comprehensive and analytic analysis on the properties of the planetary caustics . under the perturbative approximation , we derive analytic expressions for the location , size , and shape of the planetary caustics as a function of @xmath0 and @xmath2 . based on these expressions combined with those for the central caustics derived in paper i , we compare the similarities and differences between the planetary and central caustics . we provide an expression for the size ratio between the two types of caustics . we also derive an expression for the condition of the merging of the two types of caustics . we finally discuss the validity of the perturbative approximation . a planetary lensing is described by the formalism of a binary lens with a very low - mass companion . because of the very small mass ratio , planetary lensing behavior is well described by that of a single lens of the primary star for most of the event duration . however , a short - duration perturbation can occur when the source star passes the region around the caustics , which are important features of binary lensing . the caustics of binary lensing form a single or multiple closed figures where each of which is composed of concave curves ( fold caustics ) that meet at cusps . for a planetary case , there exist two sets of disconnected caustics . one ` central caustic ' is located close to the host star . the other ` planetary caustic ' is located away from the host star and its number is one or two depending on whether the planet lies outside ( @xmath3 ) or inside ( @xmath4 ) the einstein ring . the size of the caustic , which is directly proportional to the planet detection efficiency , is maximized when the planet is located in the ` lensing zone ' , which represents the range of the star - planet separation of @xmath5 . the planetary caustic is always bigger than the central caustic . we start from the formula of @xcite for the position of the planetary caustics ( eqs . [ 49 ] and [ 50 ] of his paper ) . keeping up to the first order term , the formula are expressed as @xmath6 @xmath7 where @xmath8 is a variable and @xmath9^{1/2}. \label{eq3}\ ] ] in these expressions , the coordinates are centered at the position on the star - planet axis with a separation vector from the position of the star of @xmath10 where @xmath11 is the position vector of the planet from the star normalized by @xmath1 ( see figs . [ fig : one ] and [ fig : two ] ) . the origin of the coordinates corresponds to the center of the planetary caustic . for the pair of the planets with separations @xmath0 and @xmath12 , the centers of the caustics are separated from the star by the same distance ( because @xmath13 ) but directed toward opposite directions ( because @xmath14\neq { \rm sign } [ { \bold r}(1/s)]$ ] ) . therefore , the center of the caustic is located on the same and opposite sides of the planet with respect to the position of the star for the planets with @xmath3 and @xmath4 , respectively . if one defines the lensing zone as the range of the planetary separation for which the planetary caustic is located within the einstein ring , the exact range of the lensing zone is @xmath15 to the first - order approximation , the size of the planetary caustic is proportional to @xmath16 as shown in equations  ( [ eq1 ] ) and ( [ eq2 ] ) . we will discuss the deviation of the approximation from the exact value in   4.3 .      in this case , between the two values of @xmath17 in equation  ( [ eq3 ] ) only the one with ` + ' sign is valid because the other one with ` @xmath18 ' sign results in @xmath19 . as a result , there exists only a single set of caustics for planets with @xmath3 as shown in figure  [ fig : one ] . the planetary caustic of the planet with @xmath3 is composed of four cusps , with two of them are located on the @xmath20 axis and the other two are located on the @xmath21 axis ( see fig . [ fig : one ] ) . the positions of the individual cusps , @xmath22 , corresponds to the cases of @xmath23 ( for the two cusps on the @xmath20 axis ) and @xmath24 ( for the other two cusps on the @xmath21 axis ) . then , the positions of the cusps on the @xmath20 and @xmath21 axes are expressed respectively as @xmath25 @xmath26    if we define the horizontal and vertical widths of the planetary caustic as the separations between the cusps on the individual axes ( see fig . [ fig : one ] ) , the widths are expressed respectively as @xmath27 @xmath28 where the expressions after the arrow are those evaluated to the first non - vanishing order in @xmath0 in the limiting case of @xmath29 . then , the vertical / horizontal width ratio is expressed as @xmath30 in the limiting case of @xmath29 , @xmath31 and @xmath32 , i.e. the caustic size decreases as @xmath33 and the shape becomes less elongated as the star - planet separation increases .      in this case , @xmath17 in equation  ( [ eq3 ] ) is valid only in the following range of @xmath8 @xmath34 for @xmath8 within these ranges , there are two possible values of @xmath17 corresponding to the signs . as a result , there exist two sets of caustics for planets with @xmath4 ; one above and the other below the star - planet axis ( see fig .  [ fig : two ] ) . each of the caustics for the planet with @xmath4 is composed of three cusps . one of them is located on the @xmath21 axis but the other two are not located on either of the axes . the caustic meets the @xmath21 axis at @xmath35 $ ] and @xmath36 when @xmath24 ( see fig . [ fig : two ] ) . among these two positions , the former corresponds to the cusp , and thus the location of the on - axis cusp is @xmath37 where the sign ` @xmath38 ' is for the cusps located above and below the star - planet axis , respectively . if we define the vertical width of the caustic as the separation between the two crossing points at @xmath39 and @xmath40 , the width is expressed as @xmath41 where the factor ` 1/2 ' is included into consideration that there exist two planetary caustics for planets with @xmath4 and the expression after the arrow is that evaluated to the first non - vanishing order in @xmath0 in the case of @xmath42 . by defining the center of _ each _ caustic as the midpoint between the two crossing points ( see fig . [ fig : two ] ) , its position is expressed as @xmath43    the other two cusps occurs when @xmath44 ( or @xmath45 ) . this condition is satisfied when @xmath46 ( or @xmath47 ) . then , combined with the possible range in equation  ( [ eq11 ] ) , the values of @xmath8 corresponding to the off - axis cusps are found to be @xmath48 with this value combined with equations  ( [ eq1 ] ) and ( [ eq2 ] ) , the positions of the off - axis cusps are expressed as @xmath49,\cr }   \label{eq16}\ ] ] where @xmath50 . in the limiting case of @xmath42 , equation  ( [ eq16 ] ) is approximated as @xmath51 because @xmath52 , @xmath53 , and @xmath54 . by defining the horizontal width as the separation between the two off - axis cusps , the width is expressed as @xmath55 once again , the factor ` 1/2 ' is included into consideration that there are two planetary caustics . from equations  ( [ eq13 ] ) and ( [ eq18 ] ) , the vertical / horizontal width ratio is expressed as @xmath56 in the limiting case of @xmath42 , each caustic shrinks as @xmath57 , c.f . @xmath58 for planets with @xmath3 , and @xmath59 , c.f . @xmath32 for planets with @xmath3 . based on the analytic expressions derived in the previous section , we now investigate how the properties of the planetary caustics such as the location , size , and shape vary depending on @xmath0 and @xmath2 . we also compare the properties of the planetary caustics with those of the central caustics .      in the upper panel of figure  [ fig : three ] , we present example planetary caustics of several planetary systems with different @xmath0 and @xmath2 . in the lower panel , we present the separation of the caustic from the planet - hosting star as a function of @xmath0 . in figure [ fig : four ] , we also present the variation of the caustic size ( as measured by the horizontal and vertical widths ) and the shape ( as measured by the vertical / horizontal width ratio ) as a function of @xmath0 . the properties of the planetary caustics found from the figures and the dependence of these properties on the planet parameters are as follows .    1 .   for @xmath3 , the location of the caustic center depends on @xmath0 but not on @xmath2 . on the other hand , for planets with @xmath4 , the caustic location depends on both @xmath0 and @xmath2 . in this case , the caustic is located farther away from the star - planet axis as @xmath2 increases ( see eq . [ [ eq14 ] ] ) . 2 .   although the caustic size depends on the mass ratio as @xmath60 , the shape of the caustic does not depend on @xmath2 and solely dependent on @xmath0 ( see eqs . [ [ eq10 ] ] and [ [ eq19 ] ] ) . 3 .   the rate of decrease of the caustic size with the increase of @xmath61 are different for planets with @xmath3 and @xmath4 . compared to the caustic of the planet with @xmath3 , the rate of decrease is steeper for the planet with @xmath4 . in the limiting cases of @xmath29 and @xmath42 , the caustic sizes decrease as @xmath58 and @xmath62 for planets with @xmath3 and @xmath4 , respectively ( see eqs .  [ [ eq8 ] ] , [ [ eq9 ] ] , [ [ eq13 ] ] , and [ [ eq18 ] ] ) . @xcite presented the analytic expressions for the location , cusp positions , width , and shape of the central caustics analogous to those presented in the previous section for the planetary caustics . the expressions for the location of the central caustic , analogous to equations  ( [ eq1 ] ) and ( [ eq2 ] ) for the planetary caustic , are @xmath63 @xmath64 where @xmath65 is a variable and the coordinates are centered at the position of the host star . there exists a single central caustic regardless of @xmath0 and it has an elongated asteroid shape with four cusps , of which two are located on the @xmath20 axis and the other two are off the axis . the analytic expressions for the positions of the individual cusps , which are analogous to equations  ( [ eq6 ] ) and ( [ eq7 ] ) for the planetary caustic with @xmath3 and to equations  ( [ eq12 ] ) and ( [ eq16 ] ) for the planetary caustic with @xmath4 , are @xmath66 , \label{eq22}\ ] ] @xmath67 , \label{eq23}\ ] ] where @xmath68^{1/2 }   \}$ ] . the horizontal and vertical widths of the central caustic defined as the separations between the cusps on and off the star - planet axis are expressed respectively as @xmath69 @xmath70 which are analogous to those in equations  ( [ eq8 ] ) and ( [ eq9 ] ) for the planetary caustic with @xmath3 and to equations  ( [ eq13 ] ) and ( [ eq18 ] ) for the planetary caustic with @xmath4 . then , the width ratio of the central caustic is @xmath71 which is analogous to those in equations  ( [ eq10 ] ) and ( [ eq19 ] ) for the planetary caustics with @xmath3 and @xmath4 , respectively . in the limiting cases of @xmath29 and @xmath42 , the size of the central caustic decreases respectively as @xmath72    the planetary and central caustics have the following similarities and differences .    1 . unlike the planetary caustic , the pair of the central caustics with separations @xmath0 and @xmath12 are identical as demonstrated by the fact that the inversion @xmath73 in equations  ( [ eq20 ] ) and ( [ eq21 ] ) results in the same expressions . 2 .   while the dependence of the size of the planetary caustic on the planet / star mass ratio is @xmath60 , the dependence of the central caustic is @xmath74 . therefore , the planetary caustic shrinks much more slowly with the decrease of the planet mass than the central caustic . 3 .   for planets with @xmath3 , the rate of decrease of the size of the central caustic with the increase of @xmath61 is similar to that of the planetary caustic with @xmath3 , i.e.  @xmath75 ( see eqs .  [ [ eq8 ] ] and [ [ eq27 ] ] ) , but smaller than that of the planetary caustic with @xmath4 , which shrinks as @xmath57 ( see eq . [ [ eq18 ] ] ) . then , what is the size ratio between the planetary and central caustics . if we use the horizontal width as a representative quantity for the caustic size , the size ratio between the two types of the caustics is found from equations  ( [ eq8 ] ) , ( [ eq18 ] ) , and ( [ eq24 ] ) and expressed as @xmath76 & for $ s<1 $ , \cr } \label{eq28}\ ] ] where the additional subscripts ` p ' and ` c ' denote the planetary and central caustics , respectively . in figure [ fig : five ] , we present the size ratio as a function of @xmath0 and @xmath2 . since @xmath77 while @xmath78 , the dependence of the size ratio on the mass ratio is @xmath79 . for a given mass ratio , the size ratio is maximized at around @xmath80 and decreases rapidly with the increase of @xmath61 . , the change rate of the size ratio is reversed as @xmath61 further increases beyond a critical value ( @xmath81 or @xmath82 ) . however , this reversal occurs at the separation beyond the lensing zone . ]    as @xmath83 , the location of the planetary caustic , i.e.  @xmath84 , approaches the position of the central star , around which the central caustic is located . then the two types of the caustics eventually merge together , resulting in gradual loss of distinction between the two types of caustics . the condition for the merging of the two caustics is that the separation between the two caustics is smaller than the half of the sum of the individual caustic widths , i.e.@xmath85 by using the analytic expressions for @xmath86 ( eqs . [ [ eq8 ] ] and [ [ eq18 ] ] ) and @xmath87 ( eq .  [ [ eq24 ] ] ) , we compute the region of the caustic merging in the parameter space of @xmath88 and presented in figure  [ fig : five ] ( the region enclosed by thick dashed lines ) . the region is confined in a small region around @xmath89 , but the width of the region increases as @xmath2 increases because the caustic size increases with the increase of @xmath2 .      are the presented analytic expressions based on perturbative approximation good enough for the description of the caustics in planetary microlensing ? we answer this question by comparing the two sets of caustics constructed based on analytic and numerical computations .    in figure [ fig : six ] , we present some pairs of the planetary caustics with different values of the planet parameters @xmath0 and @xmath2 . in each panel of the figure , the blue caustic is drawn by using the analytic expressions while the red caustic is the exact one based on numerical computations . for reference , we note that the mass ratios of the planets with masses equivalent to the jupiter , saturn , neptune , and earth around a host star with @xmath90 of the most probable galactic lensing event are @xmath91 , @xmath92 , @xmath93 , and @xmath94 , respectively . from the figure , we find that although the deviation increases with the increase of the planet / star mass ratio , the analytic approximation well describes the planetary caustic in most mass regime of planets ( @xmath9510^{-3}$ ] ) . for the earth - mass planet , we find that the two caustics are eventually indistinguishable . we derived analytic expressions for the location , size , and shape of the planetary caustic as a function of the star - planet separation and the planet / star mass ratio under perturbative approximation . based on these expressions , we conducted comprehensive analysis on the properties of the planetary caustics . combined with the analogous expressions for the central caustics derived in paper i , we compared the similarities and differences between the planetary and central caustics . we also presented the expressions for the size ratio between the two types of caustics and for the condition of the merging of the two types of caustics . these analytic expressions will be useful in understanding the dependence of the planetary lensing behavior on the planet parameters and thus in interpreting the planetary lensing signals . we would like to thank j.  h. an and a. gould for making helpful comments . work by c.h . was supported by the astrophysical research center for the structure and evolution of the cosmos ( arcsec ) of korea science and engineering foundation ( kosef ) through science research center ( src ) program .
although some of the properties of the caustics in planetary microlensing have been known , our understanding of them is mostly from scattered information based on numerical approaches . in this paper 
 , we conduct a comprehensive and analytic analysis of the properties of the planetary caustics , which are one of the two sets of caustics in planetary microlensing , those located away from the central star . under the perturbative approximation 
 , we derive analytic expressions for the location , size , and shape of the planetary caustic as a function of the star - planet separation and the planet / star mass ratio . based on these expressions combined with those for the central caustic , which is the other set of caustics located close to the central star , we compare the similarities and differences between the planetary and central caustics . 
 we also present the expressions for the size ratio between the two types of caustics and for the condition of the merging of the two types of caustics . 
 these analytic expressions will be useful in understanding the dependence of the planetary lensing behavior on the planet parameters and thus in interpreting the planetary lensing signals .