****The Impulse Approximation****

Transcription

1 ****The Impulse Approximation**** Elena D Onghia Harvard University edonghia@cfa.harvard.edu We consider a disk of stars on circular orbits comprising the victim interacting with an object that we will refer to as a perturber. Employ a coordinate system so that the disk is in the x-y plane. Then, the coordinate vector to any star within the victim is given by: r (t) = (x, y,z) We assume static the stars in the victim and the position vector to the perturber will be: R (t) = (X(t),Y(t),Z(t)) We adopt here the convention that the unperturbed disk always lies In the x-y plane. The acceleration of each star relative to that on the center of mass of the perturbed body is: dv dt = ψ 1 M ρ( r ') ψ( r ')d 3 r ' where ψ is the potential between the perturber and the victim, M is the mass of the victim and the integral is over the density profile of this object. For simplicity treat the force on each star in the disk from the perturber as that from a point mass. Then, the potential is:

2 GM ψ = p r (t) R (t) The velocity impulse delivered by the encounter can be obtained by integrating over time. Thus, the leading term in the series is: Δ v = GM p r R 3 R ( r R ) dt 3 R 5 We consider a coplanar case between the victim and the perturber: R (t) = (b,vt,0)

3 x[b 2 + (Vt) 2 ] 3b(xb + yvt) Δv x = GM p [b 2 + (Vt) 2 ] 5 / 2 dt r R = (xb + yvt) R 2 = b 2 + ( Vt) 2 Δv y = GM p y[b 2 + (Vt) 2 ] 3Vt(xb + yvt) [b 2 + (Vt) 2 ] 5 / 2 z[b 2 + (Vt) 2 ] Δv z = GM p [b 2 + (Vt) 2 ] 5 / 2 dt dt It is convenient to introduce: u=vt/b Δv x = GM p b 2 V [ x [1+ u 2 ] 3 / 2 3x [1+ u 2 ] 5 / 2 ]du Δv y = GM p b 2 V [ y [1+ u 2 ] 3 / 2 3yu 2 [1+ u 2 ] 5 / 2 ]du Δv z = GM p b 2 V [ z [1+ u 2 ] 3 / 2 ]du

4 Δv x = 2GM p b 2 V Δv y = 0 Δv z = 2GM p b 2 V x

5 *** Tidal Quasi-Resonance *** When the victim is spinning a resonance occurs when the spin frequency of the system matches the orbital frequency of the perturber. When the victim is a disk galaxies encountering an external perturber (e.g. an other galaxy) the resonance is responsible for the formation of long tails and bridges of stras as a result of the tidal interactions (Toomre & Toomre 1972) FIG. 1. Tail-making during a prograde encounter between two equal-mass galaxies, represented by point-particles. For clarity, the test particles comprising only one of the disks are shown. In prograde collisions, the stars in the disk are in near-resonance with the perturber during the interaction and are continuously pulled either inward or outward depending on their internal orbital phases in relation to the orbital motion of the perturber. (From Toomre & Toomre (1972)).

6 3 FIG. 2. A retrograde version of the encounter shown in Figure 1. In this case, the stars in the disk are pulled alternately inward and outward during the collision, with little net effect. (From Toomre & Toomre (1972)). As shown in the Figures above it is obvious that prograde encounters (the victim is spinning in the same direction as the perturber orbit) than retrograde ones. Mathematically, the condition for a strong resonant response is expressed by: Ω disk Ω orb v r V R 1+ e where v and r are the internal rotation velocity and size of the disk, respectively, V and R are the orbit velocity and separation and e the eccentricity of the orbit.

7 Figure 10. Resonant stripping response of the small dwarf galaxy (the victim) on progra orbits under the influence of the gravitational perturbations from the larger dwarf gala (perturber). The sequence depicted above is counterclockwise. For prograde encount the star particles in the victim (drawn as circles), which are rotating in the disk, reson and are continuously pulled either inward or outward depending on their initial positio in relation to the gravitational perturbation by the larger galaxy. As an example follow path of the blue circle in the victim.

8 Figure 11. Resonant stripping response of the small dwarf galaxy (the victim) on retro grade orbits under the influence of the gravitational perturbations from the larger dwa galaxy (perturber). The sequence is clockwise. For a retrograde collision, stars in the dis of the victim are pulled alternately inward and outward (right panel) with little net result. 4 As an example follow the path of the blue circle in the victim. The traditional impulse approximation does not distinguish between prograde and retrograde encounters therefore completely misses the resonant response.

9 Straight-line trajectory The case of a perturber moving along a straight line relative to the victim is the simplest one to analyze and is appropriate for high-speed encounters, as in clusters of galaxies. For definiteness, take the orbit path to be R(t) = (b, Vsl t, 0), (11) where, as indicated in Figure 3, b is the distance of closest approach (the impact parameter), which occurs at time t =0, and V sl is the velocity of the encounter, which is constant for a straight-line trajectory. The internal motions of the disk particles are given by eq. (2) with φ(t) =Ωt + φ 0, (12) where Ω the internal angular frequency of the victim where Ω is the internal angular frequency of the victim and ϕ 0 is the phase at the minimum distance from the perturber. Ω > 0 for prograde encounters Ω < 0 for retrograde encounters. ± We further define the non-negative parameter α b α = Ω b V sl.

10 Integrating the same equations to deliver the velocity impulse to the disk stars we arrive at: r cosφ [ 0 v x = 2GM pert b 2 αk 1 (α)+α 2( )] K 0 (α)±k 1 (α) V sl r sinφ [ 0 v y = 2GM pert b 2 α 2( )] K 0 (α) ± K 1 (α) V sl where K_0 and K_1 are the modified Bessel functions. Note that α 0 corresponds to a slowly rotating system and the expressions above reduce to the usual expressions for the impulse approximation. Comparison with numerical simulations The figures below illustrate the validity of the resonance approximations by displaying the time evolution of the victim under the effect of a perturber. Top panels illustrate the system with particle (star) velocity kicked at the starting time by an increment according to the resonance approximation. Bottom panels always display the evolution in time of the victim when a real perturber is passing close. The figures show the prograde and retrograde cases, respectively. Note the excellent agreement.

11 Prograde encounter Retrograde encounter D Onghia et al. 2010, ApJ in prep

12 Finally, notice that these considerations apply to several studies in astrophysics including star-star encounters, warps of disks and planetary systems in addition to galaxy-galaxy encouneters.