The study of two-body scatterings is a fundamental topic in physics that has been investigated for over a century. When two particles interact, the y e xchange energy and momentum, resulting in a change in their direction and speed. When such scattering events occur around a massive object, the particles can experience a gravitational deflection due to the object's gravitational field, altering their impact parameter and scattering angle.
The concept of the gravitational sphere of influence was first introduced by Laplace to study close encounters between comets and Jupiter in our Solar system. This sphere defines the region where the motion of objects is dominated by the gravity of a celestial body, and the gravity of other objects can be ignored temporarily. The radius of the sphere can be defined in several ways, such as Hill's radius and Laplace's radius, which depend solely on the masses of the objects involved.
Öpik ( Öpik 1976 ) introduced a model for calculating the trajectories of asteroids following close encounters with planets. The model assumes that encounters are instantaneous and that the gravity of the Sun can be temporarily ignored, which allows for the explicit expression of the post-encounter orbital parameters of the asteroid from two-body scattering with pre-encounter initial conditions. If the encounter is fast enough that the orbital deflection from the planet is negligible, Opik's method works well o v er a wide range of parameter space. Ho we ver, when orbital deflection from the planet is significant, ⋆ E-mail: yihan.wang@unlv.edu this method fails to accurately predict the post-encounter trajectories of the scattering objects.
Initially, Opik's method was thought to be useful only when the impact parameter between the asteroid and the planet was smaller than the radius of the sphere of influence, and the two-body scattering model became invalid when the impact parameter was too large. Ho we ver , later in vestigations (Greenberg, Carusi & Valsecchi 1988 ;Carusi, Valsecchi & Greenberg 1990 ;Valsecchi, Froeschl é & Gonczi 1997 ) found that Opik's method becomes less reliable because the pre-encounter deflection from the planet results in a different encounter geometry than originally assumed. However, if this deflection can be accurately calculated or a more solid velocity-dependent (implicitly or explicitly) sphere of influence can be used to validate the parameter space of Opik's method, accurate analytical post-encounter trajectory predictions are still possible. The goal of this paper is to identify the accurate parameter space in which Opik's method can be properly used and apply the method to several astrophysical phenomena to obtain analytical cross-sections for these events.
F ortunately, in man y cases of astrophysical interest, we are in a regime that does permit analytical solutions that are valid to high accuracy. In particular, in the case of a star with planets or protoplanets in a disc, or the analogous situation of stars or stellar-mass black holes (BHs) in an active galactic nucleus (AGN) disc, the gas disc provides a preferred orientation for orbits around the central object.
Stars and stellar-origin BHs are expected to be found in the discs of AGNs, either due to in situ formation (e.g. Stone, Metzger & Haiman 2017 ) or due to capture from the nuclear star clusters (e.g. McKernan et al. 2012 ; Kennedy et al. 2016 ; Bartos et al. 2017 ;Fabj et al. 2020 ). Once in the disc, BHs and stars are subject to frequent dynamical interactions (Samsing et al. 2022 ;Wang et al. 2021c ), and our formalism allows us to easily identify regions of parameter space in which the outcome of the scattering is especially interesting. In particular, in the following we will consider in detail two cases: one in which the scattering leads to the tidal disruption of the star by the stellar-mass BH (micro-tidal disruption event, TDE; Perets et al. 2016 ;Ya n g et al. 2020 ;Wang, Perna & Armitage 2021a ;Kremer et al. 2022 ;Ryu, Perna & Wang 2022 ), and one in which the star gets scattered within the sphere of influence (for tidal disruption) of the central supermassive BH (SMBH), hence giving rise to a standard TDE (Rees 1988 ;Evans & Kochanek 1989 ;Phinney 1989 ).
The paper is structured as follows: In Section 2, we describe the problem and the analytical solutions for the post-scattering trajectories, and present analytical formulas suitable for inclusion into N -body codes; we further test these formulas against numerical simulations. In Section 3, we apply our results to two astrophysical scenarios: an AGN disc, for which we determine the rate of TDEs and micro-TDEs, and a protoplanetary/transition disc, where we determine the distribution of free-floating and highly eccentric planets. We finally summarize and discuss the caveats of our analysis in Section 4.
To construct our formalism, we begin with free two-body scattering and we consider the turning angle after the encounter. The scattering object is assumed to be on a hyperbolic orbit with semimajor axis a hyp , eccentricity e hyp , and total energy of the two-body system > 0. If we choose one of the objects as a reference frame, r hyp , the distance of the other object to the origin, can be written as
where P = a hyp (1 -e 2 hyp ) is the semilatus rectum, and θ is the true anomaly. θ = 0 yields r = a hyp (1 -e hyp ), which corresponds to the pericentre distance. Using energy conservation, we can calculate the semimajor axis of the hyperbolic orbit. The specific total energy of the system at t = -∞ is v 2 ∞ / 2, while the total specific energy of the Keplerian orbit is -μ 12 /2 a hyp , where μ ijk ... = G ( m i + m j + m k + ...) is the gravitational constant of an N -body system. Therefore, from
2 a hyp , we get
The eccentricity of a conic section is given by
where b is the impact parameter. Using equation ( 1 ), we can calculate the corresponding θ( t ) of r hyp = +∞ ,
The turning angle of the hyperbolic trajectory
can then be written as
Plugging in the semimajor axis and eccentricity, a hyp and e hyp , respectively, we can finally express the turning angle as a function of v ∞ and b :
The time around the closest approach τ turn = t ( θ = π /2) -t ( θ = -π /2) can be calculated as
where M ( t ) is the mean anomaly (the fraction of a Keplerian orbit's period that has elapsed) of the hyperbolic orbit. This corresponds to the time that the relative distance between m 1 and m 2 is smaller than a hyp (1 -e 2 hpy ) (later on, we will show that this is consistent with the well-known Hill's sphere of influence and our experiments find that this is a good approximation for the turning time). For θ = ±π /2, the corresponding M ( t ) are
M θ =-π/ 2 = -e hyp e 2 hyp -1 + ln ( e hyp + e 2 hyp -1 ) ,
respectively. Therefore, the time-scale of the scattering event (timescale to turn) can be estimated as
The semilatus rectum P can also be rewritten as
where h is the specific angular momentum.
If the two-body scattering happens under the presence of a massive object, at the closest approach between the two light objects, the centre of mass of these will undergo a nearly Keplerian motion around the massive object. The corresponding time-scale of this motion is
where m 1 and m 2 are the masses of the light objects and m 3 is the mass of the heavier one. Comparing this time-scale with the turning timescale we obtained in the last subsection, we obtain the time-scale ratio
If the ratio τ turn / τ orb is small enough, the centre of mass mo v ements around the massive object can be safely ignored during the scattering process.
In the reference frame of the third massive body m 3 , before the scattering, if the velocity of m 1 is v 1 and the velocity of m 2 is v 2 , we can rewrite these velocities as
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The first term in each equation is the centre of mass velocity of m 1 and m 2 , while the second term represents the velocity in their centre of mass reference frame. If the condition
is satisfied, the motion of the centre of mass of m 1 and m 2 can be safely ignored during the turning time. Therefore, during this time, only the second term of equations ( 13) and ( 14) changes.
As shown in Fig. 1 , in the centre of mass reference frame of m 1 and m 2 , due to energy and angular momentum conservation, the scattering maintains the magnitude of the velocities of m 1 and m 2 but turns them by an angle of θ turn . Therefore, in the reference frame of m 3 , the velocity of m 1 and m 2 after the scattering can be expressed as
where R ( θ turn ) is the rotation matrix on the scattering plane, with
and m 12 is the total mass of m 1 and m 2 . The position vectors of m 1 and m 2 can be assumed to be unchanged (the scattering region is very small compared to r ).
The simplest scattering configuration between m 1 and m 2 in the potential of m 3 is one in which m 1 and m 2 mo v e in Keplerian orbits and encounter at a location with distance r from m 3 . In this situation, the corresponding velocities and positions of m 1 and m 2 in the m 3 reference frame are
where μ j 3 = G ( m j + M 3 ), p j and e j are the semilatus rectum and eccentricity of m j 's orbit, respectively, and ν 1 = ν 2 is the true anomaly that is obtained via cos ν j = ( p j / r -1)/ e j . E ( ω, i, ) is the Euler rotation matrix with argument periapsis ω, inclination i , and longitude of the ascending node .
Scatterings between an object ( m 2 ) in a circular orbit and one in an eccentric orbit ( m 1 ) with the same semimajor axis a 1 = a 2 = r can be used to demonstrate most of the possible outcomes of the twobody coplanar scattering around a massive body. As shown in Fig. 2 , there are eight possible scatterings in this scenario due to different orbiting directions and relative positions of m 1 and m 2 . However, because of the symmetry of the orbits, there are scatterings that result in the same post-scattered orbits. Thus, only four unique scatterings exist. We label the symmetric scatterings with the same number of labels as in Fig. 2 . The outcomes of non-coplanar scatterings can also be obtained from this framework by using the rotation matrix R ( θ turn ) on the v 1 -v 2 plane. Several degrees of non-coplanarity during scattering can result in significant changes in the outcomes, as indicated by previous studies (e.g. Rice, Rasio & Steffen 2018 ). None the less, the insights gained from our study into strict coplanarity can still be useful towards a deeper understanding of scattering processes.
To safely ignore the gravity of m 3 during the scattering process between m 1 and m 2 , the time-scale ratio τ turn / τ orb needs to be much smaller than one,
Let us first consider the case of scattering between two objects in circular orbits; we then derive
The prograde case is the well-known Hill's radius R Hill =
r, where r is the distance between m 3 and the centre of mass of ( m 1 , m 2 ). For the retrograde case, where the relative velocity between m 1 and m 2 is larger than for the prograde case, one might expect that the scattering is faster due to the larger relative velocity. Ho we ver, a hyp is much larger than the one in the retrograde case. It takes a longer time for m 1 to fly out of the region r < a hyp (1 -e 2 hpy ) to be an asymptotic straight line. Therefore, a smaller impact parameter is required for a two-body scattering approximation. Because these two cases give the extreme values of v ∞ (retrograde case gives the maximum value, while prograde case gives the minimum value), for other scatterings with non-zero eccentricities, the critical impact parameter b is bracketed in between these two extreme values.
One should note that our numerical scattering experiments indicate that in the extreme case of prograde circular scattering with zero eccentricity, m 1 and m 2 may undergo continuous resonance scatterings so that no complete single scattering can be found in the continuous scattering patterns. The ef fecti ve twobody scattering model in this paper can only be used for single scatterings in which m 1 and m 2 become unbound after the first encounter.
The post-scattered velocities of m 1 and m 2 can be obtained by using equations ( 16) and ( 17). Then, the orbital parameters of m 1 and m 2 can be calculated by
where l i = r i × v i and ǫ i ∼-μ 3 2 a i are the specific angular momentum and specific energy, respectively.
To validate the correctness of equations ( 16) and ( 17
We verify the simplest retrograde circular scattering where both m 1 and m 2 orbit around m 3 in circular orbits in the opposite direction. We also test the results for scattering locations at different distances r from the central object m 3 to verify that length in this problem can be scaled freely with the Hill radius.
For retrograde circular scatterings, the initial phase difference between m 1 and m 2 is π and the impact parameter (in this case, the semimajor axis difference) is d . F or ev ery single scattering, the simulation stops when the post-scattered relative phase between m 1 and m 2 becomes π again. We found that the semimajor axis difference may not be a good approximation of the impact parameter when d < 10 -4 R Hill as indicated by Fig. 3 . Therefore, we run a set of simulations to obtain the relationship between the semimajor axis difference and the real impact parameter. The real impact parameters are calculated from the closest approach and relative velocity between m 1 and m 2 obtained from the simulations. Indeed, when the orbital separation d is large, and hence the binding energy between m 1 and m 2 is small, then their total energy is positive and the orbit of m 2 with respect to m 1 is hyperbolic at the closest approach. Ho we ver, as d decreases, the binding energy between m 1 and m 2 becomes larger. At a critical point where the binding energy is equal to the total kinetic energy of m 1 and m 2 , the orbit becomes parabolic, which ef fecti vely makes e 12 approach 1. This leads to b approaching 0.
We do not test prograde scatterings with zero eccentricity because from the simulations we found that prograde circular scatterings result in continuous scatterings in which m 1 and m 2 continuously swap their positions after the first close approach. This makes it difficult to distinguish individual scattering between these continuous scatterings. This limitation will be discussed in the last subsection.
Fig. 4 shows, for the study case B, the comparison between the analytical approximation given by equations ( 16), ( 17 the two-body scattering for a region of impact parameters ranging from the minimum collision value (indicated by the vertical grey line) to the maximum value. The simulation results indicate that even for bigger impact parameters b > b valid, max , the analytical approximation still describes the post-scattered eccentricity and semimajor axis very accurately. The bottom panel indicates that the relative orbital energy change for case A is very weak and the total orbital energy of m 1 and m 2 is conserved during the close encounter. Fig. 5 shows the same comparison but for the larger mass set of case A. The post-scattered semimajor axis, eccentricity, and orbital energy change from the simulations fit the analytical results. The upper and bottom panels indicate that if the lighter object (i.e. m 1 ) obtains enough energy ( E / | E 0 | > 1) from the scattering, m 1 will be ejected from the system, resulting in a ne gativ e semimajor axis.
For a given closest approach R min between m 1 and m 2 , the corresponding impact parameter for this closest approach is
where v ∞ is the pre-scattering Keplerian velocity difference determined by the orbital parameters of m 1 and m 2 at the scattering location r with a 1 = a 2 = r and e 2 = 0,
III and IV .
(
Then b min can be expressed as where ± = 1 ± 1 ± e 2 1 . This expression has the following limits:
where
m 12 . This is useful to obtain the corresponding impact parameter if a given closest approach is required (i.e. collision between m 1 and m 2 , or m 1 is tidally disrupted by m 2 ).
The scattering between m 1 and m 2 can eject either m 1 or m 2 from the potential of m 3 . To calculate the critical impact b for ejecting m 1 , we can plug the Keplerian velocities into equations ( 16), ( 17 ), and ( 24) and solve e ′ = 1. Solving the general case with arbitrary e 1 is not easy. Ho we ver, it is relatively easy to get the solution in the limit of
. Similar results can be obtained in the limit of e
The abo v e conditions require the mass ratio m 2 / m 1 to be
, II and III .
(32)
The scattering between m 1 and m 2 can also lead to a direct collision with m 3 . This case requires the post-encounter angular momentum of the small object to be zero. Fig. 6 illustrates the possible solution for the post-encounter angular momentum of small objects to be zero. If the initial position vector (unchanged during the scattering) lies outside of the allowed turning region, it is impossible for the post-encounter velocity vector to be aligned with the position vector. Thus, no solution can be obtained for L ′ = 0. If there are two intersections between the position vector and the allowed turning region, there will be two solutions of b with L ′ = 0. Of course, if only one intersection can be found, then there is only one solution for b with zero post-encounter angular momentum.
The solutions for the four types of scattering are
Type I scatterings correspond to the case indicated by the black line (no solution). Type III and IV scatterings are indicated by the green line (one solution), while type II scatterings are represented by the red line (two solutions).
In this section, we show the post-encounter orbital properties for the two cases we introduced in Section 2.7 . These situations will be discussed further as astrophysical scenarios in Section 3. Fig. 7 shows the post-scattered semimajor axis and eccentricity of the lighter object m 1 in case A for the different types of scatterings I, II, III, and IV shown in Fig. 2 . This case is a good example of scattering between a main-sequence star and a stellar-mass BH around an SMBH. It indicates that type I and IV generally increase the semimajor axis of the lighter object m 1 until the impact parameter is down to 10 -4 R Hill where ejection of m 1 starts to appear. In case I, m 1 transfers energy to m 2 but gains angular momentum from m 2 . In case IV, m 1 transfers both energy and angular momentum to m 2 . For type II scatterings, the semimajor axis of m 1 typically shrinks by a maximum factor of 2 if m 1 is in an extremely eccentric orbit. In this case, m 1 obtains energy from m 2 until ejection. For type III scatterings, low-eccentricity orbits tend to increase the semimajor axis, while high-eccentricity orbits tend to decrease the semimajor axis.
For equal mass scatterings (case B), as shown in Fig. 8 , type I and II are symmetric between m 1 and m 2 in semimajor axis change, and type III and IV show similar behaviours. For type I and II scatterings, maximum semimajor axis and eccentricity are achieved at the same impact parameter b , while in type III and type IV the maximum semimajor axis and maximum eccentricity occur at different impact parameters. Interestingly, for type I and II scatterings, low initial eccentricity cases can achieve maximum semimajor axis change with much larger impact parameters, while in type III and IV scatterings, maximum semimajor axis change presents around 10 -4 R Hill for all initial eccentricities. Because the mass ratio between m 1 and m 2 is MNRAS 523, 2014-2026 (2023) 1 1 -e 2 1 m 3 m 12 1 / 3 R Hill .
unity, based on the calculation in Section 2.9 , no ejection can be obtained in the case.
In the following, we will discuss the direct astrophysical implications of our analytical results for the two cases mentioned earlier: (i) stellarmass BH-star scatterings under the potential of a central SMBH (such as, for example, the disc of an AGN); and (ii) planet-planet scattering in the potential of a host star, which is the typical situation of a planetary system with coplanar planetary orbits.
If, during the scattering of a stellar-mass BH of mass m 2 and a main-sequence star of mass m 1 and radius R * , the star gets within a distance r t = R * ( m 1 3 m 3 ) 1 / 3 of the BH, the tidal force from the BH will tidally disrupt the star, giving rise to long X-ray/gamma-ray flares (e.g. Perets et al. 2016 ).
Plugin r t as R min into Section 2.8 , we can directly obtain the critical impact parameter for a micro-TDE
The cross-section of this micro-TDE can thus be estimated via
This expression has the following limits:
where
. Note that we have constructed a cross-section with units of length, rather than area, since our scattering problems are coplanar, and hence two-dimensional (2D). Fig. 9 shows the cross-section of the micro-TDE as a function of r for the four different types of scatterings shown in Fig. 2 and for different orbital eccentricities of the star. Generally, the rate of micro-TDEs stays nearly constant if r < r c and increases as r 1/2 in the region of r > r c . In the outer region, the orbital velocity difference between m 1 and m 2 around the SMBH m 3 is significantly smaller than that in the inner region, for both prograde scatterings (I and II) and retrograde scatterings (III and IV). Thus, the gravitational focusing effect in the larger r region is stronger. Since the star will be destroyed at the fixed radius R μTDE , a stronger focusing effect in the larger r region leads to a larger cross-section of the micro-TDE. This focusing effect is extremely strong in prograde circular scatterings as shown in the upper two panels of Fig. 9 , which contributes most of the micro-TDEs in AGN discs.
Instead of being disrupted by the stellar-mass BH or the SMBH, the star could also be unbound from the SMBH by the BH during the scattering. To eject the star, the post-scattered semimajor axis as calculated from equation ( 24 ) needs to be smaller than zero. Therefore, the cross-section of the star ejection is
where L ej encompasses all the regions in the parameter space of the impact parameter b that give a post-scattered eccentricity of the star smaller than zero (i.e. the post-scattered orbit is hyperbolic). From Section 2.9 , we then obtain that the cross-section of star ejection in the limit of e 1 → 0 is
In the opposite limit of e 1 → 1, the cross-section is
R Hill , I and IV 0 , II and III .
.
Fig. 10 shows the ejection cross-section of the star for different types of scatterings and different initial orbital eccentricities of the star. For prograde scatterings in type I and II, due to the low relativ e v elocity between m 1 and m 2 in nearly circular orbits, the gravitational focusing between m 1 and m 2 can be very strong, and hence stars with low-eccentricity orbits can be easily disrupted by the BH before they acquire enough energy to be ejected. Only stars in a highly eccentric orbit, requiring less energy for ejection, can hence be ejected before the disruption. This is consistent with what we obtained from equation ( 42 ).
For retrograde scatterings in type III and IV, due to the large relativ e v elocity between m 1 and m 2 , gravitational focusing is much weaker than that for prograde scatterings. Thus, an encounter with a much smaller impact parameter can be achieved without star disruption. Therefore, even stars with low-eccentricity orbits can be ejected by the stellar-mass BH. The horizontal dashed lines show the approximation obtained from equation ( 42), indicating that the ejection cross-section is ef fecti vely independent of r . The sharp vertical cut-offs mark where micro-TDEs take o v er. In the left small r region, the required impact parameter for star ejection is smaller than the micro-TDE impact parameter. Therefore, the star will be tidally disrupted by the stellar-mass BH and hence there is no ejection.
'Standard' TDEs, in which the star is disrupted by the SMBH in quiescent galactic nuclei, have been extensively studied in the literature. The resulting flares are widely used to study the properties of the SMBH (mass and spin) as well as the populations of the host galactic nucleus (e.g. et al. 2011 ). TDEs in AGN discs are due to either (a) a 'standard' TDE in a nucleus where the TDE orbit crosses an AGN disc (Kathirgamaraju, Barniol Duran & Giannios 2017 ;Chan et al. 2019 ) or (b) a dynamical interaction between two or three bodies in the AGN disc scatters the star on to the SMBH (McKernan et al. 2022 ). 2 In case (a), the TDE is due to the standard two-body scattering into the loss cone independent of the disc and the rate of occurrence is the same as regular TDEs for that galaxy type. In case (b), the TDE is due to two-body or three-body scattering in the AGN disc and the rate of occurrence of such events is a function of disc size and the number of embedded objects within it. TDEs in AGN discs can create unique signatures because of the presence of the disc and must correspond to a source of AGN variability (Graham et al. 2017 ). Note that in the discussion below we assume that scattered stars in AGN discs are ∼1 M ⊙ . Since stars on prograde orbits within AGN discs can very rapidly grow to O (100 M ⊙ ) (Cantiello et al. 2021 ), this corresponds to assuming that the scattered star lies on an embedded retrograde orbit. In our scattering model, the scatterings between m 1 and m 2 can result in a star orbit with a very small pericentre. The SMBH could tidally disrupt the star if this enters the tidal disruption radius of the SMBH,
The cross-section for these TDEs can be obtained via
where L TDE encompasses all the regions in the parameter space of the impact parameter b that give a post-scattered star orbit with pericentre smaller than R TDE .
Solving for the post-scattered pericentre r p , 1 = a 1 (1 -e 1 ) = R TDE analytically is not straightforward. Ho we ver, as sho wn in Section 2.10 , it is relatively easy to solve for r p , 1 = 0. These solutions indicate the most probable impact parameter for TDE to happen. The TDE cross-section can be found by solving r p , 1 = a 1 (1 -e 1 ) = R TDE . Ho we v er, we hav e not been able to find an analytical solution for it. Ho we ver, we can find a scaling for the TDE cross-section. That is, σ TDE ∝ √ r . Fig. 11 shows the SMBH TDE cross-section as a function of the distance r from the SMBH. These TDEs are completely forbidden in the inner region of the AGN disc in our set-up since they require a high eccentricity of the stellar orbit. Ho we ver, this is almost impossible to obtain from the scattering between a 1 M ⊙ star and a 30 M ⊙ BH, as evinced by equations ( 16) and ( 17) and shown in Fig. 7 . SMBH TDEs start to emerge around 10 4 r g where the post-scattered eccentricity of the star orbit could reach unity. Type II and type III scatterings contribute most of the TDEs.
For planetary systems with multiple planets, the interactions between planets may lead to chaotic evolution of the planet orbits, causing planet orbits to cross. Once the planet orbits can cross each other, the scattering between planets can significantly change the architecture of the planetary system. This scattering process will last until the two planet orbits become well separated or one of the planets is ejected from the system by a very close encounter (Chatterjee et al. 2008 ;Li et al. 2014Li et al. , 2021 ; ;Pu & Lai 2021 ). In the latter scenario, the lefto v er planet is usually associated with high eccentricity (Lin & Ida 1997 ;Ford & Rasio 2008 ;Juri ć & Tremaine 2008 ;Spurzem et al. 2009 ;Li, Mustill & Davies 2020 ;Wang, Perna & Leigh 2020 ) and the ejected planet becomes a free-floating planet (Sumi et al. 2011 ; Beaug é & Nesvorn ý 2012 ) unless it is recaptured by other planetary systems. This is one of the potential mechanisms that can be used to explain the high eccentricity of exoplanets and free-floating planets. Similar to the star-BH ejection case, the cross-section of the planet ejection is well described by equations ( 39)-( 43 ).
Fig. 12 shows the cross-section of the planet ejection between an Earth-mass planet and a Jupiter-mass planet. For prograde scatterings, it is difficult to get planet ejection from circular planet orbits; only high-eccentricity orbits have the chance to be ejected from the system. The corresponding impact parameters between the two planets are roughly 10 -3 R Hill -10 -1 R Hill . For retrograde scatterings, even circular planet orbits have the chance to be ejected, although retrograde scatterings are much rare in planetary systems. Fig. 12 also indicates that the ejections happen in the outer region of the planetary system. If the scatterings are too close to the host star, planet-planet collisions will occur.
Although Fig. 8 shows equal mass and equal orbital energy scatterings (ejection never happens), we can see from the trend if we compare it with Fig. 7 that the lefto v er planet will become very eccentric once the other planet gets ejected from the system. This agrees with the general picture of planet-planet scattering in the literature.
We have presented a fully analytical solution to the two-body scattering problem in the presence of a central gravitational potential, in planar geometry. Our solution is highly accurate under the condition that the duration of the scattering event (as measured by the turning time of the scatterer's orbits) is much smaller than the orbital time in the potential of the third body. The valid parameter space in the circular scattering limit is given by equation ( 23), where for prograde scatterings, the impact parameter between two small objects needs to be smaller than the (well-known) Hill's radius, and for retrograde scatterings, the impact parameter needs to be an even smaller number ∼ ( m 12 / m 3 ) 1/6 R Hill . For other scatterings with non-zero eccentricities, the critical impact parameter b is between these two extreme values.
We tested the validity of our analytical solution via scattering experiments, and, as illustrative examples, we applied the analytical solution to compute cross-sections of astrophysical events in planar geometry under the presence of a third massive body providing the central potential. These include scattering events between stars and BHs in the SMBH potential of AGN discs, which lead to micro-TDEs, where the star is disrupted by the stellar BH it is scattering with, and SMBH TDEs, where the scattered star ends up on a very eccentric orbit that leads it to a plunge within the tidal disruption radius of the SMBH, yielding an AGN-TDE. An accurate cross-section of these events is provided in Section 3 . More generally, our analytical formulation can be very ef fecti ve at saving computational time, while obtaining accurate results, in planar two-body scatterings under the presence of a massive third body. From the calculated critical impact parameters and cross-sections, we can summarize some interesting results (due to the geometry of the 2D scattering set-ups, all crosssections listed below are in length units). (i) Micro-TDEs in AGN discs are most contributed from prograde scatterings between a star and a stellar-mass BH in low-eccentricity orbits (hundreds to thousands of geometry cross-section of the tidal radius). For retrograde scatterings, if the scattering potion is < 10 4 r g , the cross-section of the micro-TDE is basically the geometry crosssection of the tidal radius.
(ii) The cross-section of star ejection (by a stellar-mass BH) in AGN discs is roughly 10 -5 -10 -4 R Hill , ef fecti vely independent of the scattering position with respect to the SMBH, although, in the inner region of the disc, the star gets tidally disrupted instead.
(iii) TDEs by SMBH are relatively rare compared to micro-TDEs in AGN discs. The cross-section is roughly 10 -7 -10 -4 R Hill for orbits without extreme initial eccentricity. The cross-section scales with scattering position in r 1/2 . The inner region of the disc is nearly forbidden for macro-TDEs due to the micro-TDEs.
(iv) For planet-planet ejection, equal mass scatterings are more difficult to produce ejected planets than unequal mass scatterings. The cross-section for planet ejection is roughly 10 -2 -10 -1 R Hill . The inner region is also forbidden for planet ejection due to the planetplanet collision.
We emphasize that the analytical approximation we derived in the paper is meant to describe the two-body single scattering around a massi ve object. Ho we ver, for prograde circular scatterings, where the two small objects orbit around the massive object in the same direction, multiple continuous scatterings can occur. In the scattering parameter space, i.e. the space of impact parameter and relative velocity, these multiple scatterings can be hard to distinguish; e.g. the orbital turning from one scattering is not finished when the following scattering begins. The equations we derived require every single scattering to be finished; i.e. the turning process needs to be complete. Thus, equations ( 16) and ( 17) cannot properly describe close multiple scatterings for which each individual scattering cannot be discriminated. Ho we v er, via numerical e xperiments, one can identify the regions of the parameter space where resonant scattering (multiple scatterings) does not occur, so that the equations derived here can be properly used.
For the environments of AGN discs, two-body scatterings can be complicated due to the existence of the gaseous environment. These gas effects may lead to very different post-scattered results, thus significantly changing the cross-section/rate of the events we discussed in this paper. The descriptions and cross-section/rate estimates of the various astrophysical events made in this paper are all based on an assumption of no gas effects. More sophisticated computations, inclusive of gas effects, need to be performed to give more accurate cross-sections/rates of the various events.
All calculations after Section 2.8 are based on the assumption of equal energy orbit scatterings. This is the most probable situation for scatterings between two Keplerian orbits. For more general cases, another free parameter indicating the energy ratio between the two planet orbits is required in all the expressions of the calculated critical impact parameters and cross-sections.
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MNRAS 523,2014-2026 (2023)
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This picture is complicated by the very different evolution of stars embedded in AGN discs(Cantiello, Jermyn & Lin 2021 ;Jermyn et al. 2021 ) and which could have a significant impact on dynamical encounters within the disc and the disc itself(Jermyn et al. 2022 ).
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