the rayleigh - bnard convection problem is one of the fundamental problems in the physics of fluids .the basic phenomena of the rayleigh - bnard convection in horizontally extended systems are widely known .the influence of the side walls , although not studied as throughly as the horizontally extended case , is of practical importance for engineering applications .  in this paperwe study the rayleigh - bnard convection in a 2-d rectangular domain with no - slip boundary conditions for the velocity .this problem is also closely related to the problem of infinite channel with rectangular cross - section which has been studied by davies - jones , luijkx platten and kato fujimura among others .the linear aspects of the problem we consider in this paper have been studied by lee boyd , mizushima and gelfgat . in these papers , the critical rayleigh number and the structure of the critical eigenmodeshave been studied for small aspect ratio containers .  from dynamical transition and pattern formation point of view , ma and wang proved that under some general boundary conditions , the problem always undergoes a dynamic transition to an attractor as the rayleigh number crosses the first critical rayleigh number .they also proved that the bifurcated attractor , homological to , where is the number of critical eigenmodes .  in the 2-d setting that we consider , is either 1 or 2 and the latter case can only happen at the critical length scales where two modes with wave numbers and become critical simultaneously .when , the structure of is trivial which is merely a disjoint union of two attracting steady states .thus our task in this paper is to classify the structure of the attractor when .this has been studied recently in for the 3d rayleigh - bnard problem where the boundaries were assumed to be free - slip for the velocity and the wave numbers of the critical modes were assumed to be equal .the main mathematical challenge in this paper is due to the no - slip boundary conditions since the separation of variables for the linear eigenvalue problem which works in the free - slip case is not possible anymore . to overcome this difficulty ,the main approach for our study is to combine rigorous analysis and numerical computation using spectral method .as we know , spectral methods have long been used to address the hydrodynamic instability problems .in fact , in his seminal work , orszag studied the classical orr - sommerfeld linear instability problem using a chebyshev - tau method . in this paper , to treat the linear eigenvalue problem , we employ a legendre - galerkin method where compact combinations of legendre polynomials , called generalized jacobi polynomials , satisfying all the boundary conditions are used as trial functions .the main advantage of our legendre - galerkin method is that the resulting matrices are sparse which allows a very efficient and accurate solution of the linearized problem ; see also hill straughan and gheorghiu dragomirescu .once the eigenpairs of the linear problem are identified , the transition analysis is carried out by reducing the infinite dimensional system to the center manifold in the first two critical eigendirections .the coefficients of this reduced system are calculated numerically .our main results are described below .we first classify the eigenmodes into four classes according to their parities using the symmetry of the problem .then we numerically show that the first two unstable modes are always parity class one or two. then we study the transition near the critical length scales where two eigenvalues become positive simultaneously .next , we rigorously prove that the local attractor at small supercritical rayleigh numbers is in fact homeomorphic to the circle which has four or eight steady states with half of them as stable points and the rest as saddle points. the critical eigenmodes are always bifurcated steady states on the attractor and when the attractor has eight steady states , the mixed modes which are superpositions of the critical eigenmodes are also bifurcated .second , let and denote the two largest eigenvalues of the linearized problem . we find that a small neighborhood of = in the plane can be separated into several sectors with different asymptotical structures . in particular , we find that there is a critical prandtl number for the first two critical length scales and , such that for , there is a sector in this plane for which mixed modes are stable fixed points of the attractor . for ,the mixed modes are never stable and instead there is a sector in this plane in which both of the critical eigenmodes coexist as stable steady states . in this case , the initial conditions determine which one of these eigenmodes will be realized .the critical prandtl number is around for the first critical length scale and around for the second critical length scale . for higher critical length scales we found that mixed modes are never stable points of the attractor .third , recently ma wang has developed the dynamic transition theory to study transition and bifurcation problems in nonlinear sciences ; see .this paper is a first attempt to combine this theory with the numerical tools of the spectral methods to study the detailed structure of the transition and pattern formation .the paper is organized as follows . in section 2 , the governing equations and the functional setting of the problem is discussed . in section 3 ,linear eigenvalue problem is studied .section 4 states the main theorem .section 5 is devoted to the proof of the main theorem . in section 6 ,we demonstrate a method to compute the coefficients of the reduced system .and the last section discusses the results obtained by our analysis .two dimensional thermal convection with no - slip , perfectly conducting boundaries can be modeled by the boussinesq equations . the governing equations on the rectangular domain read as here is the velocity field , is the temperature field and is the pressure field .these fields represent a perturbation around the motionless state with a linear temperature profile .the dimensionless numbers are the prandtl number pr and the rayleigh number r which is also the control parameter . represents the unit vector in the z - direction .the equations are supplemented with no - slip boundary conditions for the velocity and perfectly conducting boundary conditions for the temperature .  for the functional setting ,we define the relevant function spaces : for , let and be defined by with denoting the leray projection onto the divergence - free vectors .the equations supplemented with initial conditions can be put into the following abstract ordinary differential equation :   for results concerning the existence and uniqueness of solutions of , we refer to foias , manley , and temam .  finally for , , define the following trilinear forms . first study the eigenvalue problem which reads as below we list some of the properties of this eigenvalue problem .  *the linear operator is symmetric .hence the eigenvalues are real and the eigenfunctions are orthogonal with respect to product .moreover there is a sequence such that . is found by setting in . in this casethe problem becomes an eigenvalue problem with as the eigenvalue .* we have which can be seen by computing the derivative of with respect to at . where is the ith eigenfunction . also at , by the third equation in as . plugging these into and integrating by parts, we see that * we denote the critical rayleigh number .that is  in does not depend on the prandtl number pr but only on . to see this ,one makes the change of variable so that the solution of for the eigenvalue of is independent of pr . by simplicity of the first eigenvalue ( see theorem 3.7 in ma wang ) , for almost every value of except a discrete set of values , in is 1 .the linear equations satisfy several discrete symmetries which may be found from the known non - trivial groups of continuous lie symmetries of the field equations ; see ( hydon , marques lopez blackburn ). however , for the problem we consider , it can be easily verified that the linear equations have reflection symmetries about the horizontal and vertical mid - planes of the domain .thus we can classify the solutions of the linear problem into four classes with different parities which are as defined in table [ tab : parity ] where , for example , means that is odd in the -direction and even in the -direction .           y. kato and k. fujimura , _ prediction of pattern selection due to an interaction between longitudinal rolls and transverse modes in a flow through a rectangular channel heated from below _, physical review e * 62 * ( 2000 ) , no . 1 , 601 .

Abstract: we study the rayleigh - bnard convection in a 2-d rectangular domain with no - slip boundary conditions for the velocity . the main mathematical challenge is due to the no - slip boundary conditions , since the separation of variables for the linear eigenvalue problem which works in the free - slip case is no longer possible . it is well known that as the rayleigh number crosses a critical threshold , the system bifurcates to an attractor , which is an sphere , where is the number of eigenvalues which cross zero as r crosses . the main objective of this article is to derive a full classification of the structure of this bifurcated attractor when . more precisely , we rigorously prove that when , the bifurcated attractor is homeomorphic to a one - dimensional circle consisting of exactly four or eight steady states and their connecting heteroclinic orbits . in addition , we show that the mixed modes can be stable steady states for small prandtl numbers .


nearly a century after einstein s theory of general relativity was found to correctly predict the precession of the planet mercury around the sun and the deflection of light by the sun s gravitational field , the problem of understanding orbital trajectories around very massive objects still retains interest as it relates to current astrophysical topics [ 1 ] such as the study of gravitational waves . among the numerous works on this subject , we should mention the classic publications of whittaker [ 2 ] , hagihara [ 3 ] and chandrasekhar [ 4 ] on the schwarzschild geometry , and the more recent work of levin and perez - giz [ 5 ] on the schwarzschild and kerr geometries .  in the work of chandrasekhar and that of hagihara , the orbits are classified into various types according to the roots of a certain cubic equation , while in the work of levin and perez - giz , the orbits are classified topologically by a triplet of numbers that indicate the numbers of zooms , whirls and vertices . in the work of levin and perez - giz ,the orbits were obtained by numerically integrating the integrable equations .these authors used the planet s energy and angular momentum as the principal physical parameters , and made extensive use of an effective potential for describing the schwarzschild orbits , as most studies on the topics of general relativity do .  in this paper , we first present , in section 2 , three explicit analytic expressions for the orbits in the schwarzschild geometry : one is for periodic orbits [ 6 ] ( including a special case that we call asymptotic ) , and two are terminating orbits .the explicit analytic expressions that we derive not only describe the precise features of the orbits ( periodic , precessing , non - periodic , terminating , etc . ) but also clearly indicate two physical parameters which can be used to characterize these orbits .these two dimensionless parameters are specific combinations of the following physical quantities : the total energy and angular momentum of the planet , the masses of the massive object and the planet , and , of course , the universal gravitation constant and the speed of light .these two physical dimensionless parameters were first used by one of us in ref.6 .we shall refer to these two quantities as the energy eccentricity parameter and the gravitational field parameter respectively ( or simply as the energy parameter and the field parameter ) .they will be defined in section 2 .we will use neither the common convention of setting , nor the energy and angular momentum of the planet by themselves , as the physical parameters for characterizing the orbits . with the energy parameter plotted on the horizontal axis and the field parameter plotted on the vertical axis, the parameter space for all possible orbits will be shown to be divisible into three sectors , which we call regions i , ii and ii , that have clearly defined boundaries .region i permits periodic and terminating orbits .regions ii and ii terminating orbits only .  in section 3 ,we describe region i for the orbits in greater detail .we first divide region i by lines each of which represents orbits described by elliptic functions of the same modulus .we then give a more physical division of region i which consists of nearly horizontal lines each of which represents orbits that have the same precession angle , and of bent vertical lines each of which represents orbits that have the same `` true '' eccentricity .the terminating orbits will be characterized by two parameters one of which is the angle at which the planet enters the center of the blackhole , and the other being the initial distance of the planet from the star or blackhole . in sections 4 and 5, we describe regions ii and ii in which all orbits are terminating , and we again divide region ii by curves of constant modulus each of which describes orbits with the same modulus .regions ii and ii are separated by the schwarzschild horizon .thus the grid of our `` map '' can be used to describe all possible orbits in the schwarzschild geometry in their entirety .the observational data related to a planet s orbit about some giant star or blackhole can be directly identified with a point having certain coordinates on our map , which can then be used for estimating the physical characteristics associated with the star or blackhole and that of the planet itself , assuming that the star or blackhole is not spinning very fast . for the kerr geometry , another dimensionless quantity , which is clearly the ratio of the spin angular momentum per unit mass of the blackhole to the orbital angular momentum per unit mass of the planet , should enter into the consideration . in section 6 ,we briefly discuss a possible extension of our results to the case involving a slowly spinning blackhole , at least to the first order perturbation , by rescaling the physical parameters involved .  in section 7 , we study the deflection of light by the gravitational field of a very massive object .a single dimensionless parameter will be used to characterize the region .we show that here too , we should divide the region into three sectors , which we again call regions i , ii and ii , and we present three analytic expressions for the trajectories of light applicable in these different regions .region i has trajectories of light that get deflected , and regions ii and ii have trajectories of light that are absorbed by and terminate at the blackhole .  in section 8 , we give a summary of our results .proofs of many interesting analytic relations among the parameters appearing in these studies are given in several appendices . since our results presented in this paper cover gravitational fields of all ranges , from the weak field produced by the sun of our solar system , for example , to the very strong field produced by a blackhole, we want to avoid referring to the massive object that produces the gravitational field as a blackhole , and prefer to refer to it as the star or blackhole , and we shall refer to the object of a much smaller mass that orbits around it as the planet or the particle .we have supplemented our many analytic results with numerous tables that present various physical quantities such as the minimum and maximum distances of the planet from the star and the angles of precession of the orbits that are calculated from our analytic expressions , as well as numerous figures that show various kinds of orbits of the planet and various kinds of deflection of light .we consider the schwarzschild geometry , i.e. the static spherically symmetric gravitational field in the empty space surrounding some massive spherical object such as a star or a blackhole of mass .the schwarzschild metric for the empty spacetime outside a spherical body in the spherical coordinates is [ 1 ]    where    is known as the schwarzschild radius , is the universal gravitation constant , and is the speed of light . if =(t , r,\theta , \phi ) ] , , , and substituting these into eq.(25 ) gives the well known approximate result given by eq.(15 ) .for these periodic orbits , we note that the maximum distance ( the aphelion ) of the planet from the star or blackhole and the minimum distance ( the perihelion ) of the planet from the star or blackhole , or their corresponding dimensionless forms  and  are obtained from eq.(20 ) when and when respectively , and they are given by    and    where and are determined from eqs.(23 ) , ( 24 ) and ( 10 ) in terms of and .although we call the orbits given by this solution for periodic , they are not necessarily closed orbits .it is seen from the precession discussed above that for , unless is a rational number , the orbit will not close and it is not strictly a closed periodic orbit .however , for all practical purposes , any irrational number when truncated becomes a rational number , and thus the orbit will be closed .the distinction of closed and non - closed orbits depending on whether is rational or irrational is of course of profound theoretical interest [ 5 ] .  for a general periodic orbit that precesses, the general or true eccentricity of the orbit is defined by    where and are given by eqs.(26 ) and ( 27 ) .we shall show in the following section that the true eccentricity is in general not equal to the energy eccentricity parameter defined by eq.(11 ) , but that in the limit of , i.e. in the newtonian limit .for the special case of however , we shall show that it coincides with the special case of for all applicable values of , and that it signifies an unbounded orbit .we now proceed to present the second solution .solution ( a2 ) for , .we write the right - hand side of eq.(9 ) as and eq.(9 ) can be integrated with expressed in term of the inverse jacobian function [ 7 ] .the equation for the orbit is found to be    where , , , and are given by eqs.(21)-(24 ) as in the first solution .this solution gives a terminating orbit .the point at has been chosen to be given by    the planet , starting from the polar angle at a distance from the blackhole , plunges into the center of the blackhole when its polar angle is given by , i.e. when    where and are given by eqs.(21 ) and ( 22 ) .the region of where orbits given by solutions a1 and a2 are applicable will be called region i , and it will be described in greater detail in section 3 . thus each point of parameter space in region i represents two distinct orbits , one periodic and one terminating . at the same coordinate point ,the characteristic quantities that describe the two distinct orbits are related .for example , by noting and from eqs.(26 ) and ( 27 ) , can be expressed as    where and are the minimum and maximum distances for the periodic orbit at the same coordinate points .it will be noted that is less than , i.e. for the terminating orbit the planet is assumed initially to be closer to the blackhole than the for the associated periodic orbit , except at where and the planet has a circular instead of a terminating orbit that will be explained later .we note that the terminating orbit equation ( 29 ) presented has no singularity at the schwarzschild horizon , because , as is well known , is a coordinate singularity and not a physical singularity .the orbit obtained from continuing beyond the value may become interesting if the concept of `` whitehole '' turns out to be of physical relevance .  for now ,the orbits in region i are characterized mathematically by .we now present the third solution .solution ( b ) for .define    where and are defined by eqs.(10 ) and ( 17 ) .the real root of the cubic equation ( 16 ) is given by    and the two complex conjugate roots and are .we further define  ^{1/4}\ ] ]  and    writing the right - hand side of eq.(9 ) , with , as , eq.(9 ) can be integrated with expressed in terms of the inverse jacobian function [ 7 ] .we find the equation for the orbit to be    this is a terminating orbit .the initial distance of the planet at has been chosen to be given by    it plunges into the center of the blackhole when its polar angle is given by    where and are given by eqs.(34 ) and ( 35 ) .again , we note that the orbit equation ( 36 ) has no singularity at .the region of where orbits given by eq.(36 ) are applicable will be divided into two sectors called regions ii and ii , the boundary between which will be defined later .they have terminating orbits only . for now , the orbits in regions ii and ii are characterized mathematically by .  as for the initial points of the orbits discussed above , by comparing eq.(19 ) with the orbit equations ( 20 ) , ( 29 ) and ( 36 ) , and with eqs.(26 ) , ( 30 ) and ( 37 ) , we already noted that our choice of in our orbit equations is such that it gives , , and respectively that in turn give , , and as the initial distances of the planet from the star or blackhole .we then note from eq.(9 ) that and hence for the planet at these initial points of the trajectories , i.e. the trajectory or more precisely the tangent to the trajectory at is perpendicular to the line joining the planet to the star or blackhole .all this will be seen in the figures presented later , and all our references to the initial position of the planet from here onward assume that the trajectory ( as increases from ) of the planet at its initial position is perpendicular to the line joining the planet to the star or blackhole .consider the orbits expressed by eqs.(20 ) and ( 29 ) given by solutions a1 and a2 and characterized mathematically by .we call the region covered by the associated range of values for region i.  to gain a preliminary perspective , consider the earth ( as the planet ) and the sun ( as the star ) in our solar system . substituting the mass of the sun and the angular momentum of the earth per unit mass of the earth , we find .the energy eccentricity parameter , which is equal to the true eccentricity of the earth s orbit for such a very small value , is known to be about .the approximate relation gives the squared modulus of the elliptic functions that describe the earth s orbit to be .we see that for the planetary system that is familiar to us , the values of and are very small indeed .we may also note that the schwarzschild radius  would be well inside the sun which has a radius of .the earth s dimensionless distance is from the sun s center . forthis value of , with , the orbit given by eq.(29 ) from solution a2 would require the initial position of a planet to be according to eq.(31 ) , i.e. the planet would have to be at a distance equal to the schwarzschild radius from the center of the sun for it to have a terminating orbit which plunges to the center of the sun . therefore the terminating orbit given by eq.(29 ) is inapplicable for our solar system .the periodic orbits , on the other hand , are perfectly valid .however , for cases when the massive object is a gigantic mass concentrated in a small radius such as a blackhole , all the possibilities presented here may arise . as the field parameter increases from , the modulus of the elliptic functions that describe the planet s orbits also increases . from eqs.(21)-(24 ), it is seen that several steps are needed to relate to and . in appendixa , we show that a direct relationship between and and can be established , and it is given by    the of eq.(39 ) is the same that appears in eq.(24 ) , and , in particular , it is equal to for respectively .  the curve represented by , after setting in eq.(39 ) , can be readily shown to give a quadratic equation that gives    for , and for .equation ( 40 ) representing gives the upper boundary ( for the values of ) of region i ( the uppermost heavy solid line in fig.1 ) ; it extends from for to for , i.e. a line that is nearly parallel to the -axis .thus region i is a region bounded by , and by where is given by eq.(40 ) , in which the squared modulus of the elliptic functions that describe the orbits cover the entire range .we now use eq.(38 ) to give a plot of lines of constant as shown in fig.1 .these lines conveniently divide region i into regions of increasing field strengths as increases from to . on a point representing a particular and a particular value , can be determined from eq.(38 ) and the orbit is then given by eq.(20 ) using eqs.(98 ) , ( 10 ) and ( 21 ) .the values of on these constant lines for the values of are given in table 1 which thus give the coordinates of the points on the lines representing different values of .these coordinate points from table 1 are used to give the following tables : tables 2 and 3 give the values of and for the orbits obtained from eqs.(26 ) and ( 27 ) .note that the dimensionless distance is in units of the schwarzschild radius which depends on the mass of the star or blackhole corresponding to that particular coordinate point , and thus one should not compare at two different coordinate points just by their absolute values alone .table 4 presents the values of the precession angle in units of , i.e. , obtained from eq.(25 ) .table 5 presents the values of the true eccentricity obtained from eq.(28 ) .tables 2 - 5 are to be used in conjunction with table 1 for identifying the locations of the corresponding quantities that are presented .the physical quantities presented in tables 2 - 5 together with the coordinates given in table 1 now give all possible periodic orbits in the schwarzschild geometry in its entirety .that is , the coordinates of a planet orbiting a non - spinning blackhole can be identified if the observation data on and can be collected .region i shown in fig.1 is where orbits given by eqs.(20 ) and ( 29 ) apply . in sections 4 and 5, we shall discuss regions ii and ii which are shown above region i in fig.2 where orbits given by eq.(36 ) apply .as an example of application of tables 1 - 5 , from the second row and second column of tables 1 - 5 and using only two significant figures , for orbits with we find from tables 2 - 5 that or and , i.e. orbits with those seemingly small values of and give a precession angle of per revolution that is already very large compared to those encountered in our solar system for which the precession angle is only per century for the earth s orbit ( for which ) , and the value of the true eccentricity of these orbits is already quite different from their energy parameter .we thus appreciate that the range of values for given by for region i , where ranges from for to for , is not as small as it seems ( noting also that ) , and that the classical newtonian orbits are restricted to a very narrow strip of the region indeed for which and and for which for .  although the lines of constant in region i conveniently associate the orbits with the orbit equations for the periodic and terminating orbits given by eqs.(20 ) and ( 29 ) and with the physical parameters given in tables 2 - 5 , the precession angle and the true eccentricity are more physically meaningful parameters that can be associated with the description of the orbit .the expressions for and in terms of and are given by eq.(99 ) in appendix a and eq.(100 ) in appendix b. for a given value of and of , we can use eqs.(99 ) and ( 38 ) to solve for ( and ) ( using a numerical program such as fsolve in maple ) and thus locate its coordinate ; and similarly for a given value of and of , we can use eqs.(100 ) and ( 38 ) to solve for ( and ) .the relationship of and with is simpler for and will be discussed later ( see eqs.(49 ) and ( 50 ) ) . in fig.3 , we present lines of constant ( that are nearly horizontal ) and lines of constant ( that are bent vertical ) in region i , and the corresponding tables for their coordinates are presented in tables 6 and 7 .we note that because given by eq.(25 ) depends on given by eq.(21 ) as well as on , the line of constant does not coincide with the line of constant except for .we note also that the line of constant does not coincide with the ( vertical ) line of constant except for .we show in appendix b that it is only for a very thin strip of region , where is between zero and some very small positive value , that which applies in the newtonian limit .we also show in appendix b that when exactly .the distinction between defined by eq.(11 ) or eq.(13 ) with defined by eq.(28 ) in the newtonian or non - newtonian theory has never been clearly recognized previously .  with fig.3 which has curves of constant and constant in place , regioni is now partitioned into cells with the coordinate points specified by .we have a clear idea what the orbits of a planet would be like at points within each cell in terms of their precession angle and true eccentricity , and the coordinates of these orbits then give the energy and field parameters corresponding to these orbits . in fig.4 , we present examples of periodic and unbounded orbits , plotted in polar coordinates , corresponding to various precession angles of ( vertically from top to bottom ) for values of ( horizontally from left to right ) , where the star or blackhole is located at the origin .we first note that the orbits for which are periodic and closed because is a rational number in for each one of them .the precession angle can be seen from the heavy solid line that marks the trajectory ( as increases ) from the initial point at to the first point at which the distance from the origin is equal to the distance at .the true eccentricity of the orbits is given by eq.(28 ) .for example , for the orbit of fig.4 ( a1 ) for , , is far from zero which can be seen from the and in the figure , and it can be more accurately calculated to be equal to . for each of the unbounded orbits characterized by , the incoming trajectory coming from infinity at makes an angle with the outgoing trajectory going to infinity given by from eq.(25 ) , as can be seen in some of the figures presented .the case corresponding to the special case of will be discussed later in this section for which the planet starting from ends up circling the blackhole with a radius that approaches ( see fig.6d ) .  generally ,if we are given a coordinate point in fig.3 , for example , a point on just above the line slightly to the left of the curve ( where and ) , then we find degrees or , and part of the orbit is shown in fig.5 .whether the orbit will close on itself depends on whether is or is not a rational number in principle , even though , as we mentioned before , a truncated number in practice is always a rational number and the orbit will be a closed one .we only show part of the orbit in fig.5 as the subsequent path is clear from the angle of precession and true eccentricity of the orbit and we are not concerned with how many `` leaves '' the orbit is going to create .figure 3 ( or one with even more curves of constant and constant ) is a very useful map that can be used fruitfully with any observation data that are obtained for any planet .  besides the special case , the case of is also somewhat special in that it allows many relationships to be expressed simply and explicitly .we present some of these simple relations for in appendix c. it is to be noted from fig.1 that the line of constant is very close to the boundary given by .the line of constant for region ii , on the other hand , is closer to dividing the region approximately into two halves , as shown in fig.2 .the curve for region ii will be discussed in section 4 .the terminating orbits in region i given by eq.(29 ) can be characterized by the planet s initial position given by eq.(31 ) , and by the angle at which the planet enters the center of the blackhole .it is interesting to note that even for these terminating orbits , the precession angle still has an `` extended '' meaning and use that we shall describe .it is clear from eq.(29 ) that the orbit terminates , i.e. becomes zero when , but if the orbit is continued ( by continuing to increase ) , would assume its initial value at when , producing a `` precession angle '' of which is equal to the precession angle for the corresponding periodic orbit at the same coordinate point .since , the polar angle at which the path of the terminating orbit enters the center of the blackhole is related simply to the precession angle of the periodic orbit by , or    as can be easily calculated from for the periodic orbits given in table 4 , we do not tabulate it separately .the values of are presented in table 8 , and we note the small range for the entire region i. examples of these terminating orbits are presented in fig.6 . the dotted line represents the continuation of the orbit when is continued beyond .before we discuss regions ii and ii , we want to describe three special cases : the case of which , as we shall see , is not of any interest but must be included for completeness ; the case of which gives the upper boundary of region i ( and lower boundary of region ii ) ; and the case of which gives the right boundary of region i ( and of regions ii and ii ) ( see figs.1 and 2 ) .\(i ) the special case of   the line of coincides with the axis in fig.1 . to show this, we note that implies from eq.(95 ) . substituting into eq.(24 )gives when we use the expressions in eq.(10 ) for and .the we find and , and from eq.(98 ) , we find    equation ( 20 ) then gives or , i.e. it is the limiting case of zero gravitational field .as we pointed out earlier , the classical newtonian case is given by only a very narrow strip represented by and for which is large but finite .\(ii ) the special case of   it follows from eqs.(38 ) and ( 39 ) that on the line of , . thus from eqs.(24 ) and( 17 ) , we have    which can be identified as the `` boundary '' between solutions a and b in section 2 .the range of values for is for ( see the discussion below eq.(40 ) ) , and for that range of values , or and therefore ( see eq.(10 ) ) . from eq.(41 ) , the relation between and can be more precisely expressed as  {g_{3}}=-\sqrt{\frac{g_{2}}{3}}\ ] ]  after noting that is negative and is positive for the values of along the line .also from eq.(98 ) , we note that    the periodic orbits given by eq.(20 ) become    where    and where the values of ( and ) are those given by the values of and on the line that are obtained from eq.(40 ) .the orbit is not a periodic orbit ; it is what we call an asymptotic periodic orbit .the planet starts from an initial position at given by    and ends up at circling the star or blackhole with a radius that asymptotically approaches given by    equations ( 43)-(46 ) are explicit and simple equations that give the orbit equation , , and for .in particular , it is seen from table 3 , for example , that ranges from for to for , i.e. is still no less than twice the schwarzschild radius for the strongest gravitational field that permits the periodic orbits .however , it is a very small number indeed compared to , say , for the earth s orbit around the sun .  on this upper boundary of region i ,the terminating orbit given by eq.(29 ) from solution a2 becomes a circular orbit with a radius  , where is the initial distance of the planet from the star or blackhole given by eq.(30 ) . from eqs.(30 ) and ( 31 ) and noting that for , we find that    given by eq.(46 ) ( see tables 3 and 8 for ) .we shall refer to the orbits given by eqs.(43 ) and ( 47 ) as the asymptotic periodic and the asymptotic terminating orbits respectively of region i. thus the special cases given by eqs.(43 ) and ( 47 ) for of the periodic and terminating orbits given by eqs.(20 ) and ( 29 ) for solutions a1 and a2 respectively clearly exhibit completely different behaviors from their counterparts for .examples of asymptotic periodic orbits are shown in fig.4 g .asymptotic terminating orbits are simply circles of radius equal to , as shown in fig.6(d ) .  using eqs.(28 ) ,( 42 ) , ( 45 ) and ( 46 ) , for the true eccentricity can be shown to be expressible in terms of by    which can be solved to give in terms of , and then in terms of using eq.(38 ) .we find that the coordinates of a given on the line are given by    and    it is interesting that eqs.(49 ) and ( 50 ) can be used in place of eq.(40 ) as parametric equations for determining the coordinates of the line as takes the values from to . in particular , eqs.(49 ) and ( 50 ) allow us to see that the curves are not vertical ( except for ) , and they intersect the upper boundary of region i for ( see fig.3 ) .the curve , the boundary curve , and the -axis are concurrent at , .we can conclude that periodic orbits with have true eccentricity in the range . as another exampleit can be shown using eqs.(49 ) and ( 50 ) that periodic orbits with have .\(iii ) the special case of   the maximum or boundary value for which is is also a special case of interest . from its definition given by eq.(11 ) , since , can not be greater than . in appendix b, we show that always gives an unbounded orbit for the periodic and the asymptotic periodic orbits of region i and the terminating orbits of region ii , but not the asymptotic terminating orbit of region i which has a radius given by eq.(47 ) independent of and thus is not an unbounded orbit .thus is the boundary for for region i ( as well as for region ii ) .many explicitly simple relationships among etc . have been found on the boundary line , and they are given and proved in appendix b. in particular , we have , on in region i , that        and    examples of unbounded orbits for are shown in fig.4 ( a3-g3 ) .we shall now describe regions ii and ii for the orbit equation ( 36 ) given by solution b for the case .consider the orbits expressed by eq.(36 ) given by solution b and characterized mathematically by .the associated values for in this case satisfy , where is the upper boundary of region i given by eq.(40 ) .this region of parameter space defined by can be naturally divided into two sectors which we call region ii and ii with region ii bordering region i ( see fig.2 ) .the boundary between regions ii and ii is determined by the schwarzschild radius in a manner to be described later in this section .  we first want to prove that the lower boundary ( for ) of region ii , characterized by as it is for the upper boundary of region i , also gives , where is calculated from eq.(35 ) for solution b ( in section 3 , we showed that for calculated from eq.(22 ) for solution a , implies ) . substituting into eq.(32 )gives  {g_{3}}=-\frac{1}{2}\sqrt{\frac{g_{2}}{3}}.\ ] ]  after noting that is a negative value for the range of values for , substituting the above into eq.(35 ) gives .we also find from eq.(34 ) that {g_{3}}/2=\sqrt{3g_{2}}/2 ] , where is some approximation constant , such that the results we have presented for the orbits in the schwarzschild geometry are approximately applicable for the orbits in the kerr geometry in terms of the scaled parameters .that is , the orbits in the equatorial plane and their characterization for the schwarzschild and kerr geometries are qualitatively very similar to the first order in except that the basic parameters , and have to be slightly rescaled .levin and perez - giz [ 5 ] obtained their orbits in the kerr geometry from numerically integrating eqs.(66)-(68 ) and it would be interesting to study and examine when and how the planet s orbits in the kerr geometry that they obtained can be related with our results with the scaled parameters , and when and how they begin to differ significantly from those in the schwarzschild geometry that we presented in this paper .we now consider the deflection of light by a gravitational field .we can not use the proper time as a parameter .so we use some affine parameter along the geodesic [ 1 ] . considering motion in the equatorial plane ,the geodesic equations give eqs.(3 ) and ( 5 ) , and we replace the -equation ( 4 ) by the first integral of the null geodesic equation , and we have [ 1 ]        where the derivative represents . substituting eqs.(69 ) and( 71 ) into ( 70 ) gives the combined energy equation    substituting and into the combined energy equation gives the differential equation for the trajectories of light in the presence of a gravitational field    the constants and have a physical significance through their ratio as follows : let denote the distance of the light beam to the center of a star or blackhole when the trajectory of the light beam is such that . can either be associated with the distance of closest approach of the light beam to the blackhole or with the initial distance to the blackhole of the light beam .the latter case is associated with light trajectories that terminate at the blackhole . with defined and letting , we can set to be equal to [ 8 ] .it is again convenient to consider the problem in terms of the dimensionless inverse distance defined by     defined here is slightly different from the defined by eqs.(8 ) and ( 19 ) previously . in terms of of eq.(74 ) , eq.(73 ) becomes    since at , one root which we call    of the cubic equation is known , and the term on the right hand side of eq.(75 ) can be replaced by , and the other two roots of the cubic equation can be found from solving a quadratic equation .we denote the three roots of the cubic equation by . thus writing eq.(75 ) as   trajectory of light represented by an equation for as a function of the polar angle obtained from integrating eq.(77 ) can be characterized by a single parameter which essentially specifies either the distance of the closest approach or the initial distance of the light beam to the blackhole ( these distances are scaled by the schwarzschild radius of the blackhole ) . as in our discussion of the planets ,our references to the initial position of the light beam assume that the trajectory of the light beam at that initial position is perpendicular to the line joining that position to the star or blackhole .the range of is clearly between and , where means that the light beam is infinitely far away from the blackhole , means that the light beam is at the schwarzschild radius at its closest approach or its initial position , and means that the light beam is at the center of the blackhole . as we show in the following ,the region can be appropriately divided into three sectors which we again call regions i , ii and ii. the similarity between the characterization of these three regions with that for the planetary orbits discussed in the previous sections will become apparent .not surprisingly perhaps , only a single parameter which we choose to be , is needed for the characterization of the trajectories of a light beam in contrast to the two parameters ( which we choose to be and ) , which we needed for the characterization of the orbits of a planet .the relationship between and , from eqs.(76 ) and ( 2 ) , is    region i : or   here denotes the distance of closest approach of a light beam that comes from a great distance .we let   ,  \notag \\e_{2 } & = & u_{1 } ,  \notag \\e_{3 } & = & \frac{1}{2}[1-u_{1}-(1 + 2u_{1}-3u_{1}^{2})^{1/2}],\end{aligned}\ ] ]  with and we consider the region , and write eq.(77 ) as    equation ( 79 ) can be integrated [ 7 ] with expressed in terms of an inverse function . after a little algebra and re - arrangement, we find the trajectory s equation in terms of the jacobian elliptic functions of modulus to be    where    the angle of deflection can be obtained as follows .if we set and also set as the incoming angle in eq.(80 ) ( see fig.9a for the special case of ) where denotes the total angle of deflection of light by the mass , we get the following equation for determining exactly :   = \frac{% ( e_{1}-e_{3})e_{2}}{(e_{2}-e_{3})e_{1}},\ ] ]  where are given by eqs.(78 ) and ( 81 ) .it can also be expressed as    where   ^{1/2}\ ] ]  equations ( 80)-(82 ) were first given by one of us in ref.6 .examples of these trajectories obtained from eq.(80 ) are presented in polar coordinates in fig.9 , where the blackhole is located at the origin . by setting the angles of deflection presented in fig.9 to be , the corresponding values of can be determined from eqs.(82 ) and ( 78 ) using the maple fsolve program , and they are found to correspond to the distances of closest approach , , , respectively .the case of is interesting as it corresponds to the light ray being turned around by . that the upper boundary of region i characterized by or is a very special casecan be seen mathematically because it results in and hence , and from eqs.(81 ) and ( 80 ) .physically , it results in the light circling the blackhole with a radius even though the trajectory has been shown to be an unstable one [ 1 ] .this known result can also be simply obtained from the equation of motion for and thus .if one compares the size of the unstable circular photon orbit with the allowed limiting radii of the planetary asymptotic periodic orbits or , one can see that the radius of the asymptotic circular path of a planet around a blackhole is still a little larger than that for a photon , but not by much .the lower boundary of region i characterized by or gives , and , and thus gives or which is a limiting case as the light ray that is infinitely far away at its closest approach to the blackhole is completely undeflected .as in the case of the region i particle orbits discussed in section 3 , the squared modulus of the elliptic functions that describe the trajectories of light here also covers the entire range ; it varies from at the lower boundary to at the upper boundary .  for small ,the trajectory of light given by eq.(80 ) has been shown [ 6 ] to reduce to    and the total deflection of light to reduce to the well known result    it can be shown from our exact result given by eq.(82 ) that this approximate expression ( 84 ) still gives an accuracy of two significant figures for or .as approaches , or as approaches , we may let , where is a small positive number . from eqs.(78 ) and ( 81 ) , we can express the quantities , and appearing in eq.(82 ) in power series in and find , to the first order in , , , and .substituting these into eq.(82 ) immediately gives an expression for which is correct to the first order in .if an attempt is made to find an expansion of near , since ^{1/2} ] , and the order of the three roots must be changed to maintain the inequality .we write   ,  \notag \\e_{3 } & = & \frac{1}{2}[1-u_{1}-(1 + 2u_{1}-3u_{1}^{2})^{1/2}].\end{aligned}\ ] ]  we consider the region , and write eq.(77 ) as    equation ( 86 ) can be integrated [ 7 ] with expressed in terms of an inverse function .after some rearrangement , we find    where and are calculated using the same expressions given by eq.(81 ) but with given by eq.(85 ) .the expressions for given by eqs.(78 ) and ( 85 ) coincide at for which , and both equations ( 80 ) and ( 87 ) give or independent of .eq.(87 ) gives a trajectory of light which terminates at the blackhole when .as in our discussion of the terminating orbits for the planet , the terminating light ray trajectories can be characterized by the angle with which the light beam enters the center of the blackhole .  as increases from to , covers the entire range ; it decreases from to . when , i.e. when the light beam grazes the schwarzschild horizon , , and we have the trajectory of light given by    which gives , for or , and for or , i.e. the light is absorbed at the center of the blackhole .examples of the trajectories of light obtained from eqs.(87 ) and ( 88 ) for ( ) and ( ) in region ii are shown as the solid lines in fig.10 ( a ) and ( b ) .the path that emerges from the center of the blackhole when is continued beyond ( shown as a dotted line in fig.10 ) again may be interesting if the concept of whitehole is of any physical relevance .when the distance to the blackhole at is inside the schwarzschild horizon , the terminating path takes on a somewhat different form as we show below .region ii : or   here has the same meaning as that in region ii . as increases beyond , i.e. when is less than the schwarzschild radius , in eq.(85 ) remains real while and become complex .we now write the three roots of the cubic equation as , and given by   ,  \notag \\ \overline{b } & = & \frac{1}{2}[1-u_{1}-i(3u_{1}^{2}-2u_{1}-1)^{1/2}].\end{aligned}\ ] ]  we consider the region , and write eq.(77 ) as    this equation can be integrated [ 7 ] with expressed in terms of an inverse function . after a little algebra ,we find    where  ^{1/4}\ ] ]  and    equation ( 91 ) gives the trajectory of light when is inside the schwarzschild horizon and it terminates at the blackhole when , where and are given by eqs.(93 ) and ( 92 ) . on the boundary with regionii where , and , from eqs.(93 ) and ( 92 ) , eq.(91 ) becomes eq.(88 ) and thus there is no discontinuity in the orbit as it makes a transition from region ii to region ii across .we note that as in the case of region ii for the planetary orbits , region ii for light trajectories covers a semi - infinite range of the parameter characterizing it ( ) but is of relevance only to a very small physical region for the initial position of a light beam inside the schwarzschild horizon .the terminating orbits of light rays in region ii are also of very little variety as is restricted to a limited range of .an example of a terminating trajectory obtained from eq.(91 ) is shown as the solid line in fig.10(c ) for ( ) ; the dotted line again represents a trajectory of light coming out from the center of the blackhole as is continued beyond .it may be of some mathematical interest to note that as or , the squared modulus of the jacobian elliptic functions used to describe the trajectories approaches a value  that is the same as that given in section 5 for the case of region ii for the planetary orbits .thus the squared modulus of the elliptic functions that describe the terminating light trajectories in region ii is restricted to a very small range even as region ii consists of a very large interval .we have presented exact analytic expressions given by eqs.(20 ) , ( 29 ) and ( 36 ) for the planetary orbits in the schwarzschild geometry .the equations relate the distance of the planet from the star or blackhole to the polar angle and are described explicitly by jacobian elliptic functions of modulus .equation ( 20 ) gives periodic orbits that describe a planet precessing around a star or blackhole , while eqs.(29 ) and ( 36 ) give terminating orbits that describe a planet plunging into the center of a blackhole .one of the most important aspects of our analysis is the construction of a map with coordinates that we use to view all possible orbits in their entirety , where the two dimensionless parameters and are defined by eqs.(11 ) and ( 12 ) which we call the energy and field parameters respectively . for , we show that there are three regions which we call regions i , ii and ii where these orbits are applicable , and where and that depend on are given by eqs.(40 ) and ( 57 ) respectively ( fig.2 ) .region i has periodic and terminating orbits given by eqs.(20 ) and ( 29 ) .regions ii and ii have terminating orbits only given by eq.(36 ) .we have divided region i into grids that consist of lines of constant precession angle given by eq.(25 ) and lines of constant true eccentricity defined by eq.(28 ) ( fig.3 ) ; the lines of constant are obtained from solving eqs.(38 ) and ( 99 ) , and those of constant from solving eqs.(38 ) and ( 100 ) .these grids make the identification of all possible periodic orbits , including the unbounded and asymptotic ones , convenient and precise . numerous numerical results for orbits in regioni are presented in tables 1 - 6 , and examples of precessing orbits , including the unbounded ones , are shown in figs.4 and 5 . among the interesting results , for example ,table 6 for  and fig.4 f3 show that a planet coming from infinity at zero polar angle , makes a complete loop about the blackhole , and returns to infinity with a polar angle that approaches .figure 3 , or a more refined version of it that can be constructed using the expressions for and that we presented , can be fruitfully used with the experimental observation data . as for the terminating orbits ,our orbit equations assume the planet s initial trajectory to be perpendicular to the line joining it to the star or blackhole .the terminating orbits of region i require the planet to be at a distance from the star or blackhole , and those of region ii requires , i.e. they require the planet to be initially very close to the blackhole or within the schwarzschild horizon .the terminating orbits of region ii , on the other hand , are more interesting as the planet can be initially at a distance .we have shown how the periodic orbits of region i become the asymptotic periodic orbits as , and then become terminating orbits as becomes greater than .it is interesting to note that a planet initially at infinity with and in a trajectory perpendicular to the line joining it to a blackhole can be absorbed by the blackhole if is greater than .we have also presented exact analytic expressions given by eqs.(80 ) , ( 87 ) and ( 91 ) for the trajectories of light in the presence of a star or blackhole depending on the value of one parameter that has a range which can be divided into three regions : regions i , ii and ii , where is defined by eq.(76 ) . in regioni , the deflection of light can range from small angles to going continuously around the star or blackhole in a circle . in regionsii and ii , light is absorbed into the center of the blackhole . among the interesting results, a deflection of light by requires a distance of closest approach to the blackhole equal to ( fig.9b ) , and for , light will be absorbed by the blackhole .we have thus presented a complete map that can help to identify characteristics of stars and blackholes ( that are not spinning too fast ) from the observed characteristics of objects or light beams that are affected by them .* electronic address : fhioe.edu  [ 1 ] m.p .hobson , g. efstathiou and a.n .lasenby : general relativity , cambridge university press , 2006 , chapters 9 and 10 .  [ 2 ] e.t .whittaker : a treatise on the analytical dynamics of particles and rigid bodies , 4th edition , dover , new york , 1944 , chapter xv .[ 3 ] y. hagihara , japanese journal of astronomy and geophysics viii , 67 ( 1931 ) .[ 4 ] s. chandrasekhar : the mathematical theory of black holes , oxford university press , 1983 , chapter 3 .[ 5 ] j. levin and g. perez - giz , phys .d 77 , 103005 ( 2008 ) .[ 6 ] f.t .hioe , phys .lett . a 373 , 1506 ( 2009 ) .[ 7 ] p.f .byrd and m.d .friedman : handbook of elliptic integrals for engineers and scientists , 2nd edition , springer - verlag , new york , 1971 .see in particular p.72 , 74 , 81 , 86 for the formulas used in this paper .  [ 8 ] see e.g. j. l. martin : general relativity : a guide to its consequences for gravity and cosmology , ellis horwood ltd ., chichester,1988 , chapter 4 .[ 9 ] s.v .iyer and a.o .petters , gen .relat . and grav .39 , 1563 ( 2007 ) , c.r .keeton and a.o .petters , phys .d 72 , 104006 ( 2005 ) .in this appendix , we derive the relation among and given by eqs.(38 ) and ( 39 ) . substituting eq.(23 ) into eq.(22 ) and after a little algebra , we find    and hence we find    we then find    equations ( 96)-(98 ) and ( 23 ) allow to be expressed in terms of and as  substituting eq.(96 ) into the relation , and substituting the result into eq.(24 ) gives the relation among and given by eqs.(38 ) and ( 39 ) . for , we get from eqs.(38 ) and ( 39 ) , and from eq.(98 ) we get .  from eqs.(25 ) , ( 21 ) , ( 98 ) and ( 10 ) , we find an expression for the precession angle given by eq.(25 ) in terms of and to be   this appendix , we show the relation between the energy parameter and the true eccentricity .the energy parameter in all three regions i , ii and ii is defined by eq.(11 ) .the general or true eccentricity is defined by eq.(28 ) . from eq.(28 ) , ( 26 ) and ( 27 ) and using the expressions ( 98 ) of in terms of given in appendix a , we find that can be expressed as    or    for small and , we find that    because for very small and [ 6 ] .we thus confirm the identification of defined in eq.(11 ) with the eccentricity of the orbit in newtonian mechanics . as we pointed out in the text , is generally not equal to .however , as we show below , exactly when and in this case the orbits are unbounded .  for the possibility of unbounded orbits in region i , we set the initial at in eq.(20 ) , and get    or    using eqs.(23 ) , ( 10 ) and ( 97 ) that give in terms of and , and after a little algebra , we find the simple equation that relates to on to be given by eq.(51 ) .this simple equation ( 51 ) between and for can be used for any .for example , we find that for , , and for , . for , and for small , we have which is a special case of that is valid more generally for .  in addition, we find $ ] and therefore    and from the expression for and given by eq.(98 ) , we find    thus from the expressions ( 27 ) , ( 30 ) , and ( 21 ) for , , and , we find that when , they have the simple expressions given by eqs.(53 ) , ( 54 ) and ( 52 ).  also , substituting eq.(48 ) into eq.(100 ) shows that when , i.e. and coincide at .it is known that in elliptic functions , the squared modulus is a special value for which many simple relations arise .we first consider the case of in region i. we note that substituting into eq.(22 ) gives a relation , and substituting the expressions of from eq.(23 ) into this relation gives , or .thus which results in    from eq.(24 ) , which in turn gives a simple relationship    for example , we have for , and for ( using lhospital rule ) .these two values represent the two terminal coordinates of the constant line in region i ( see fig.1 ) .the other solution of eq.(106 ) is eq.(58 ) which is applicable for region ii as we shall show later in this appendix .we also find from eq.(23 ) that    and  {g_{2}},\ ] ]  and the orbit equation ( 20 ) becomes    where the value for in this case is given by eq.(107 ) for region i. the precession angle can be found from eq.(25 ) and from .it is given by  {g_{2}}}-2.\ ] ]  from eqs.(26)-(28 ) and ( 108 ) , we find    which can be inverted and solved for in terms of , giving    substituting eq.(111 ) into eq.(106 ) gives in terms of as    since for , when as we showed above , substituting this value into eq.(106 ) gives . for , when , and substituting this value into eq.(106 ) gives as it should .thus eqs.(111 ) and ( 112 ) are the parametric equations for the line of constant which can be used instead of eq.(107 ) as takes the values between and .  also , substituting from eq.(108 ) into eq.(27 )gives    independent of for , as shown in table 3 .we now consider in region ii . from eqs.(35 ) and ( 33 ) , we have    from eq.(32 ) , and from , and , we arrive again at eq.(106 ) with the same given by eq.(10 ) .this explains why we stated after eq.(107 ) that the other solution of eq.(106 ) given by eq.(58 ) gives the relation between and for region ii .the two equations ( 108 ) and ( 58 ) that give simple relations between and in two different regions and that arise as two different solutions of the same equation ( 106 ) show a rather remarkable symmetry exhibited by the special case .it also follows from eqs.(32 ) and ( 34 ) that     ^{1/4},\ ] ]  and that the orbit equation ( 36 ) becomes    where the value for the above equations is given by eq.(58 ) for .  since ,the initial distance of the planet from the blackhole is from eq.(37 ) , independent of , as shown in table 10 .on the upper boundary of region ii , the planet starts from the schwarzschild horizon given by , which implies that for , from eq.(36 ) , or                  finally , there is a special case of or ( ) in region ii that is somewhat interesting . if of eq.(10 ) is equal to zero , then from eq.(17 ) , and from eqs.(32)-(35 ) , we find , , , , and noting that for and , we have   which is independent of , i.e. the constant curve in region ii just above the boundary curve of regions i and ii in fig.2 is a horizontal line . thus the terminating orbits represented by eq.(36 ) for any point along this horizontal line are represented by elliptic functions of the same squared modulus given above .it should be remembered , however , that the orbits and the initial positions of the planet ( the initial trajectory of which is perpendicular to the line joining it to the blackhole ) depend also on and that are dependent on the value of .thus the orbit for , for example , that gives ( and ) is a terminating orbit for a planet that is initially infinitely far away from a blackhole .

Abstract: exact analytic expressions for planetary orbits and light trajectories in the schwarzschild geometry are presented . a new parameter space is used to characterize all possible planetary orbits . different regions in this parameter space can be associated with different characteristics of the orbits . the boundaries for these regions are clearly defined . observational data can be directly associated with points in the regions . a possible extension of these considerations with an additional parameter for the case of kerr geometry is briefly discussed .   pacs numbers : 04.20.jb , 02.90.+p  = 1


stochastic volatility models such as the heston model have become one of the standard approaches in financial option pricing .for some stochastic volatility models and under additional restrictions , closed - form solutions can be obtained by fourier methods ( e.g. , ) .another approach is to derive approximate analytic expressions , see e.g. and the literature cited therein . in general , however , even in the heston model when the parameters in it are non constant the partial differential equations arising from stochastic volatility models have to be solved numerically .  in the mathematical literature, there are many papers on numerical methods for option pricing , mostly addressing the one - dimensional case of a single risk factor and using standard , second order finite difference methods ( see , e.g. , and the references therein ) .more recently , high - order finite difference schemes ( fourth order in space ) were proposed that use a compact stencil ( three points in space ) . in the option pricing context , see e.g.  .  there are less works considering numerical methods for option pricing in stochastic volatility models , i.e. , for two spatial dimensions .finite difference approaches that are used are often standard , second - order methods , e.g. in where different efficient methods for solving the american option pricing problem for the heston model are proposed . in a high - order compact finite difference scheme for option pricing in the heston model is derived and this approach is extended to non - uniform grids in .other approaches include finite element - finite volume , multigrid , sparse wavelet , fft - based or spectral methods .the classical alternating direction implicit ( adi ) method , introduced by peaceman and rachford , douglas , fairweather and mitchell , is a very powerful method that is especially useful for solving parabolic equations ( _ without _ mixed derivative terms ) on rectangular domains .beam and warming , however , have shown that no simple adi scheme involving only discrete solutions at time levels and can be second - order accurate in time in the presence of mixed derivatives . to overcome this limitation , unconditionally stable adi schemes which are second order in timehave been proposed by hundsdorfer and verwer and more recently by int hout and welfert .these schemes are second - order accurate in time and space . in different second - order adi schemes of this typeare applied to the heston model . in approach is combined with different high - order discretisations in space , using high - order compact schemes for two - dimensional convection - diffusion problems _ with mixed derivatives and constant coefficients_. in this approach is combined with sparse grids and applied to multi - dimensional diffusion equations , again _ with constant coefficients_. building on the ideas in , a high - order ( second - order accurate in time and fourth - order accurate in space ) adi method for option pricing in stochastic volatility models which involve the solution of two - dimensional convection - diffusion equations _ with mixed derivative terms and space - dependent coefficients _ is derived in .  in this chapterwe combine the approaches from and , to obtain a _ sparse grid high - order adi scheme _ for option pricing in stochastic volatility models . in the next section we recall stochastic volatility models for option pricing and the related convection - diffusion partial differential equations .section [ sec : hv ] is devoted to the hundsdorfer - verwer adi splitting in time .the spatial discretisation is introduced in section [ sec : hoc ] for the implicit steps , and in section [ sec : hoexpl ] for the explicit steps .the solution of the resulting scheme and the discretisation of boundary conditions are discussed in sections [ sec : hosolve ] and [ sec : bc ] .the sparse grid combination technique is explained in section [ sec : ct ] .we present numerical convergence results in section [ sec : num ] .we consider the following class of stochastic volatility models : assume that asset spot price and variance follow two stochastic diffusive processes for  ] :     in order to discretise the problem and solve numerically , we truncate our spatial boundaries to finite values .take , where , and , where , so that the spatial domain forms a closed rectangle in of points with uniform spacing of in the -direction and in the -direction : the lower -boundary is truncated to to ensure non - degeneracy of the partial differential equation for all values of .we assume cell aspect ratios to be moderate .we also take a uniform partition of  ] and  ] and  ] .the grid parameters of the computational domain are chosen to be , , and . the parameters of the adi method are and , cf .section [ sec : hv ] .the full grid solution is computed with step sizes , and with , while the sparse grid solution is constructed according to definition ( [ eq : ct ] ) . in order to avoid instabilities due to the extreme distortion of the grid we neglect grids within the combination technique , where for .thus , the finest resolution in one of the sub - grids along one coordinate direction is given by .we compare the performance of the high - order adi scheme in the full and sparse grid case for a european put option with the parameters given in table [ tab : para ] .figure [ fig : experiment1 ] shows the maximum error plotted versus the grid resolution for both cases .the fourth - order compact finite difference scheme achieves an estimated numerical convergence order of , the error of the sparse grid solution decays slightly slower due to the logarithmic factor in ( [ eq : error_ct ] ) .  to illustrate the computational efficiency we compare the run - time to the accuracy in figure [ fig : experiment2 ] for both approaches .we confirm that , as the mesh width decreases , the lower number of employed grid nodes in the sparse grid method outweighs its slightly lower convergence rate .the serial implementation of the combination technique outperforms the full grid solver in the high accuracy region , reducing the computational time by about an order of magnitude , while achieving a similar accuracy .bd acknowledges support by the leverhulme trust research project grant ` novel discretisations for higher - order nonlinear pde ' ( rpg-2015 - 69 ) .ch is supported by the european union in the fp7-people-2012-itn program under grant agreement number 304617 ( fp7 marie curie action , project multi - itn strike - novel methods in computational finance ) .further ch acknowledges partial support from the bilateral german - spanish project hipeca high performance calibration and computation in finance , programme acciones conjuntas hispano - alemanas financed by daad .jm has been supported in part by a studentship under the epsrc doctoral training partnership ( dtp ) scheme ( grant number ep / l505109/1 ) .b. dring , m. fourni and a. rigal .high - order adi schemes for convection - diffusion equations with mixed derivative terms . in : _spectral and high order methods for partial differential equations - icosahom12 _ , m. azaez et al .( eds . ) , pp . 217226 , lecture notes in computational science and engineering 95 , springer , berlin , heidelberg , 2013 .                     a.r .mitchell and g. fairweather , improved forms of the alternating direction methods of douglas , peaceman , and rachford for solving parabolic and elliptic equations , _ numer ._ * 6 * , 285292 , 1964 .

Abstract: we present a sparse grid high - order alternating direction implicit ( adi ) scheme for option pricing in stochastic volatility models . the scheme is second - order in time and fourth - order in space . numerical experiments confirm the computational efficiency gains achieved by the sparse grid combination technique .


a second - order correlation function is an intensity - intensity correlation function , having information on both photon statistics and dynamics of the light generation process of a light source .it was first introduced by hanbury brown and twiss in order to measure the angular separation of binary stars  , and later it was applied to property measurement of various light sources such as measuring the coherence time of thermal light  , getting information on the nature of scatterers  and surveying the correlation properties of laser light near laser threshold  .more recently , it was used in probing the nonclassical nature of light such as antibunching  and sub - poissonian photon statistics  .much effort has been made in devising a precise and efficient apparatus to measure the second - order correlation function .the first successful time - resolved measurement was done by using a single detector , a single variable delay generator and a coincident circuit to measure the coherence time of low - pressure gas discharge in a single isotope  .this technique has a limitation in getting the correct correlation function near zero time delay due to the imperfectness of the detector such as spurious multiple pulse emission and incapability of detection for a finite amount of time just after detecting real photons .these effects are referred to as after - pulsing and dead time effects , respectively .  to overcome this limitation ,a two - detector configuration has been adopted in which a light beam is divided into two parts and two photodetectors are used to record photons arrived at each detector  . unlike the single - detector configuration , the spuriously emitted photons and dead photons at one detector are completely uncorrelated to the photons detected on the other detector so that the contribution of after - pulsing and dead time effects are equally spread over the entire measurement time .this allows measurements to be extended down to the zero time delay .more sophisticated correlators , which made use of multiple time delays , were developed which digitized the time interval between a _ start _ pulse from one photodetector and multiple _ stop _ pulses from the other photodetector at a time  .the number of pulse pairs corresponding to a given delay is registered on a corresponding counter and the result is proportional to the second - order correlation function . such a device is called a multi - stop time - to - digital converter ( mstdc ) .this method is more efficient than previous method owing to the multiple delay generators .even though a two - detector configuration can effectively remove the artifacts caused by the detector imperfectness , this benefit is true only for a stationary source , the intensity of which is independent of time . since the probability of spurious emissions or losing photons due to the detector dead time depends on the intensity , the measured intensity profile can be greatly distorted for a non - stationary source .in other words , the spurious emissions or lost photons are not equally spread over the entire measurement time but dependent on time .this can cause distortion in the second - order correlation measurement .  in the present study, we have carefully investigated the limitation of the two - detector configuration for non - stationary sources and have devised a systematic way of calibrating the second - order correlation function for the first time to our knowledge . for this study , we have developed a novel second - order correlation measurement system which records all the arrival times of photons detected at start and stop detectors and make use of all the photons detected at the start detector as triggers .there are no waste of photons at the start detector and thus our system can be much more efficient than mstdc for a light source having a high photon flux and a long coherence time .a schematic of our second - order correlation measurement system is shown in fig .[ setup ] .photons are detected by two avalanche photodiodes ( apds , model spcm - aqr-13 by perkinelmer ) .one detector ( apd1 ) serves as a start detector while the other ( apd2 ) as a stop detector .each apd has a dark count rate less than 150 cps and a dead time of about 50 ns and generates electrical pulses whenever the photons are detected with detection efficiency of about 50 .apds are electrically connected to the counter / timing boards ( counter 1 , 2 ) installed in two computers ( computer 1 , 2 ) .relatively low - cost and commercially available counter / timing boards ( pci-6602 by national instruments ) are used to record the arrival times of the electrical pulses and to store them to the computers .each board has its own internal clock of 80 mhz so that the time resolution is 12.5ns .two independent boards installed in separate computers are used to prevent the crosstalks between boards or computers .the internal clock of each counter / timing board has a limited accuracy and also has a drift of 50 ppm as the surrounding temperature changes .the accurate frequency of the internal clock in each board has been calibrated by counting the arrival times of reference pulses from function generator(ds345 , stanford research system ) .the frequency difference between boards is typically several tens of ppm and is accounted for in getting absolute arrival times from the measured ones .    to make the two counters start to count at the same time , an additional control computer ( computer 0 )is used to generate a trigger signal to simultaneously arm the counters .it has a board with analog outputs and digital inputs / outputs ( ni6703 , national instruments ) , which can send a ttl signal as a trigger .all the arrival times of photons detected on both detectors are the relative times with respect to the same origin defined by the trigger .all the detected photons on one detector therefore can be used as start pulses with respect to those of the other detector .for this reason , we call our apparatus a _ multi - start multi - stop time - to - digital converter _ ( mmtdc ) compared to the conventional _ multi - stop time - to - digital converter _ ( mstdc ).  mmtdc makes use of all the photons detected at a start detector whereas the mstdc makes use of a single photon detected at a start detector as a trigger and measures the relative arrival times for a time interval which is several times ( let us say times ) longer than a correlation time of a light source .it starts over and repeats the next measurement cycle using another single photon detected at the start detector after .if the incoming photon flux to the start detector is , only one photon out of photons is used in the measurement .therefore , our mmtdc has an efficiency times higher than that of mstdc .mmtdc is specially advantageous for a light source which has a high photon flux and long correlation time but has a limited operation time with its second - order correlation signal embedded in a large background .the microlaser  was a good example to fit this category .it had an output photon flux of about 3 mcps and a correlation time of about 10 such that our new method was about 30 times more efficient than that of mstdc .because of limited oven life time , full time measurement could give a signal - to - noise ratio of about 3 even when mmtdc was used .we could have obtained a signal - to - noise ratio of only 0.55 if we had used the conventional mstdc method , only to fail to resolve the signal . due to limitation in computer memory ,the number of arrival times recordable at a time is limited by about one million counts in our mmtdc setup .to get an enough number of data , measurements should be done in a sequential way .all computers are connected by ethernet connections in such a way that they can send and receive messages among themselves . the counting computers ( computer 1 and 2 )send a message to the control computer to notify the end of counting whenever they complete a specified number of counting and recording .after checking that the control computer has received the message , both counting computers prepare a next measurement .when the control computer receives the message , it sends triggers to the counting computers to initiate counting again .the number of sequences is determined so as to get an enough signal - to - noise ratio .  to get the second - order correlation function ,a histogram is constructed for the time differences between all possible pairs made of one of the photon arrival times at start apd and another one at stop apd . to save calculation time , only the pairs of photons the time difference of which are within a certain time window , typically chosen 10 times larger than the correlation time of the source ,are included in the calculation .the second - order correlation function can then be obtained from the normalization of this histogram by the averaged histogram value for much longer delay times than the correlation time .we have tested our mmtdc system using an extended cavity diode laser ( 2010 m , newport ) operating far above threshold as a test source .the photon statistics of its output is supposedly poissonian and the second - order correlation function is thus unity for all delay times .photon flux was 3 mcps for each detector and the number of sequences was 1,000 with 500 kilo counts per sequence .  a correlation function obtained from the photons measured at a _ single _ detector exhibits the effect of detector dead time and after - pulsing .a figure [ deadtimedistribution](a ) shows a measured result .two types of dip below 1 appear and the one near the zero time delay corresponds to the detector dead time .this can be confirmed from the output pulse shape of apd shown in fig .[ deadtimedistribution](c ) .the full width of the pulse was measured to be about 50 ns .the stop photons following a given start photon within this time window are completely ignored and thus a dip with a depth of unity appears .another dip is extended to 250 ns and its depth is about 0.25 , which means that the following stop photons are partially ignored with a probability of 25% .this dip originates from the counter / timing board .usually counter electronics also have dead times since it takes a finite amount of time to record measured arrival times . in our case , since there is no onboard memory in the counter / timing board , it has to transmit the arrival times to the computer memory through dma ( direct memory access ) .since the data transfer rate through dma is limited by 100 mbps , some counts can be missed if the time interval between photons is too short to be transferred . the probability to miss a count depends on the time interval between successive photons .we call this loss of counts the _ dead time effect of the counter _ as an analogy to the detector dead time .  the correlation function obtained from the photons detected at _ two _ detectors is shown in fig .[ deadtimedistribution](b ) .the effect of the detector and counter dead time is completely removed and the value of the correlation was very close to unity for all delay times , as expected .since the number of photon pairs per bin amounted to , the standard deviation from unity was only . utilizing all the photons detected at start detector helped to reduce the background noise greatly .using our mmtdc second - order correlation measurement system , we have measured the second - order correlation function for a non - stationary light source .the output beam from an extended cavity diode laser was modulated by an acousto - optic modulator .the amplitude of a driving rf ( radio frequency ) field to the acousto - optic modulator was sinusoidally modulated using a function generator so that the intensity of the first - order diffracted beam was sinusoidally modulated .its functional form can be written as , where and have the units of count per second for the photon counting measurement . since the response time of aom is measured to be 130 ns , the modulation frequency can be safely set up to 1 mhz . for the present experiments ,the modulation frequency was set as 100 khz .the normalized second - order correlation function for a classical source , applied to the intensity of electromagnetic field , is given by where denotes a time average .the second - order correlation function for the sinusoidally modulated source is thus calculated to be figure [ eliminationdip](a ) and ( b ) show measured second - order correlation functions .mean count rates for individual detectors were about 0.6 mcps and 1000 sequences of measurements were done with each sequence counting 300 kilo counts for each detector .it took about 1000 seconds including sequencing procedure .the adjacent points were added up and thus the time resolution was 125 ns .figure [ eliminationdip](a ) is the second - order correlation function obtained from photons detected at a single detector and thus shows a sharp dip near time delay zero .the dip results from the dead times of the detector and the counter . on the other hand , fig.[eliminationdip](b ) is obtained from the photons detected on two detectors , apd1 and apd2 .the central dip is completely eliminated , as in the case of stationary light sources .the normalized shot noise is only about 0.06% due to the large mean counts per bin of about 2.8 million .the contrast ratios of single- and two - detector configurations are almost the same .even though the two - detector configuration can eliminate the detector dead time effect near the zero time delay , the detector dead time still can affect the correlation measurements for non - stationary sources when the detector counting rate becomes comparable to the inverse of the detection dead time .since the probability to miss photons is dependent on the intensity , the measured time - varying intensity profile can be distorted .we used photon counting method to measure the time - dependent intensity .since our mmtdc was based on the photon counting method , we used photon counting technique rather than photo current measurement for a consistent quantitative relation between the intensity profile and the second - order correlation function . in order to obtain enough number of counts per counting bin for intensity measurement , which has to be done in a fashion of single shot measurement , we used a slow intensity modulation frequency of 0.1 hz for a given incoming photon flux of around 1 mhz .photons were counted for every counting bin of 0.01 sec and the counted numbers were transferred at the end of each bin . since the dead time of the counter occurs when the time interval between successive data transfers is shorter than 250ns , this measurement should be free from the counter dead time effect . at 1 mhz counting rate ,the mean counts per bin was counts and thus the normalized noise was 0.01 , which means we could resolve an intensity modulation whose contrast was as small as 0.01 .we repeated the measurement for various mean count rates while keeping the contrast of the intensity constant .to do so , we fixed the radio frequency field driving the acousto - optic modulator and varied the mean intensity using neutral density filters .the intensities measured in this way were well fitted by sinusoidal functions as shown in fig .[ contrastintensity](a ) . the contrast ratio in fig .[ contrastintensity](a ) is about , which corresponds to the contrast ratio at a mean intensity of 3 mcps in fig .[ contrastintensity](b ) , where the measured results denoted by circular dots show a linear decrease as the mean count rate increases .this linear decrease can be explained in terms of the correction factor of the detector which needs to be multiplied by the measured intensity to get an original intensity , where and are the original and measured intensities , respectively , in the unit of cps and is the dead time of the detector .if there exists additional dead times such as the dead time of the counting board , should be replaced with an effective total dead time including all dead times , and this subject will be discussed in details in the next section . for and , \;,\end{aligned}\ ] ] which shows that the contrast ratio is modified from to .note that the unmodified contrast , fixed in the experiment , can be determined from the intercept of a linear fit to the measured contrast ratios and the inclination of the linear fit corresponds to the dead time . in fig[contrastintensity](b ) , we get and 56 ns , which is about 10% larger than our initial estimate based on the single - photon pulse shape in fig.[deadtimedistribution](b ) . this discrepancy is due to the finite bin size of 12.5 ns , which results in an additional broadening of about 6 ns , a half of the bin size , in the effect of the detector dead time in the correlation function .( 100 khz ) , circular dots : expected from the measured contrasts of the intensity ( 0.1 hz modulation ) .a solid and dashed lines are theoretical results with the effective dead time of 80 ns and 56 ns , respectively.,width=316 ]  since the intensity profile is distorted by the detector dead time , the second - order correlation function will be also affected . for a measured intensity contrast of ,the expected contrast of the second - order correlation function is .they are denoted by circular dots in fig .[ g2contrast ] . for an unmodified contrast ,the dependence of the measured contrast on the mean intensity is given by .the contrast of is thus expected to be ^ 2\simeq ( 1 - 2t_d b)a^2/2b^2 \label{g2-contrast}\ ] ] the dashed line in fig .[ g2contrast ] indicates the expected contrast with ns .  using mmtdc , we have measured the second - order correlation function under the condition identical to the one under which we had measured the intensity except for the intensity modulation frequency , which is now set at 100khz , since the correlation measurement need not be done in a single - shot fashion as in the intensity measurement .the square dots in fig .[ g2contrast ] are the contrasts obtained by fitting the measured with a sinusoidal function .it decreases linearly as the photon flux increases , but the decreasing rate of the contrast ratio is 1.54 times larger than that expected from the intensity measurement .this discrepancy is due to the dead time of the counter / timing board , which was absent in the preceding intensity measurement performed at 0.1 hz modulation frequency with a data transfer rate of 100hz . in the second - order correlation measurement , regardless of the modulation frequency , since all of the arrival times are recorded , the data transfer rate can be much faster than the inverse of 250 ns , which is the maximum dead time of the counter . when the time difference of two successive photons is shorter than 250 ns , there exists a finite chance that the following photon is ignored .a complexity arises since this chance is not alway unity. it can be anywhere between 0 and 1 .we thus need to find an effective dead time which can properly includes both detector and counter dead time effects .to understand the dead time effect of counter , we numerically simulated the effect of a partial dead time .a poissonian light source was simulated using a random number generation algorithm . in the simulation , if the time difference between two successive photons are shorter than 50 ns , the following photon is omitted with a probability of 50% . this probability would be unity for the case of the detector dead time .figure [ partialdeadtime](a ) is the second - order correlation function of this simulated source and shows a dip with a half width the same as the dead time and a depth equal to , 50% .the correction factor is calculated as a function of the mean intensity and shown in fig .[ partialdeadtime](b ) .the result can be well fitted by the following relation . that the linearity coefficient in with respect to the intensity is not as in eq .( [ correctionfactor ] ) but .we can generalize this observation and expect that the detector dead time in eq .( [ g2-contrast ] ) will be replaced by . this expectation has been confirmed by our numerical simulations .therefore , we call an effective dead time and it is equal to the area of the dip ( only ) around the origin in the second - order correlation function obtained in the single - detector configuration ( fig . [ partialdeadtime](a ) ) . in general , we can experimentally determine the effective dead time of an entire detection system from the dead time distribution , _i.e. _ , the shape of the second - order correlation function around the origin , calculated with photons detected on a single detector only .    the dead time distribution of our mmtdc system is shown in fig .[ deadtimedistribution](a ) .the total area of the dip near the origin is 80 ns , which is our effective dead time for the entire measurement system .the area of the central dip only equals =55 ns , originating from the detector dead time .the ratio is 1.45 , and thus we expect that the inclination of the observed contrast in with respect to the counting rate would be 1.45 times greater than that expected from the intensity measurement free from the counter dead time effect .this is what we have observed in fig[g2contrast ] , where the observed ratio is 1.54 , showing only 6% discrepancy from our expectation .  from these observations, we can establish a calibration method for the second - order correlation function of a non - stationary light source .we first measure the second - order correlation function using the single detector configuration and determine the effective dead time of the entire system from the dead time distribution around the origin .the second - order correlation function is then measured again , but this time in the two detector configuration . since the correction factor in eq .( 6 ) is independent of the modulation frequency , we can calibrate the measured correlation function by deviding its amplitude by , where is the mean count rate .

Abstract: a novel high - throughput second - order - correlation measurement system is developed which records and makes use of all the arrival times of photons detected at both start and stop detectors . this system is suitable particularly for a light source having a high photon flux and a long coherence time since it is more efficient than conventional methods by an amount equal to the product of the count rate and the correlation time of the light source . we have used this system in carefully investigating the dead time effects of detectors and photon counters on the second - order correlation function in the two - detector configuration . for a non - stationary light source , distortion of original signal was observed at high photon flux . a systematic way of calibrating the second - order correlation function has been devised by introducing a concept of an effective dead time of the entire measurement system .