When the Sky Falls: Tidal Heating and Tidal Stripping. Lesson 4

Transcription

1 When the Sky Falls: Tidal Heating and Tidal Stripping Lesson 4 Guillermo Haro International School 2005

2 The Dynamics behind accretion and merging Last time we talked about dynamical friction and tidal forces Dynamical friction is a force that opposes the motion of an object moving in a background of particles. Its effect is bigger the more massive the moving object. Its effect is directly proportional to the background mass density. For objects a lot more massive than the background particles, the mass of the background particles does not matter. For slow velocities (compared to the mean velocity of background particles), its effect grows with the velocity of the moving object. However, for fast velocities, its effect decreases with velocity. The tidal force that limits the extent of a system within the potential well of another, depends on this size of the former and the gradient of the force exerted by the latter. Its effect can produce a radial stretching, no effect, or radial squeezing, depending on whether the density profile of the system exerting the tide is steeper, equal, or shallower than ρ r -1. Although truncation is a function of position and velocity, a good enough working definition of a boundary in configuration space can be found. It is usually taken as the Jacobi radius, which relates the average densities of truncated and truncating systems. 2

3 The Dynamics behind accretion and merging This time we will talk about tidal dissipation, tidal heating and tidal stripping, which are consequences of dynamical friction and tidal forces 3

4 The Dynamics behind accretion and merging Dynamical friction and tides, dissipate the orbital energy of an incoming system when it impinges on a target system, perturbing the internal motion of the latter. For the incoming satellite it acts as an energy sink, so it is sometimes called tidal dissipation. For the target system it acts as a source of energy, so it is sometimes called tidal heating. This heating leads to mass loss which is called tidal stripping. Tidal dissipation and tidal heating are two sides of the same coin, namely the dissipation of center of mass kinetic energy of an incoming satellite in the thermal bath provided by the larger system. Ordered kinetic energy Disordered thermal energy 4

5 The Dynamics behind accretion and merging When a fast moving, massive object moves within a background of lighter particles there are several effects that should be considered: The finite time of the perturbation. As the perturber passes through the target system, the stars inside the target move along their orbits. The target system alters the motion of the perturber. In particular, the back reaction of the perturbation on the perturber produces a non-linear coupling. Except for very idealized systems, this problem can not be solved analytically and we must turn to numerical simulations. In fact, a lack of sufficient work in this area, due in good part to the vastness of the parameter space to explore, has resulted in poor knowledge about the outcomes of galaxy-galaxy interactions, like, for instance, merging cross sections for galaxies. We need to build up statistics. 5

6 The Impulsive Approximation However, in the limit of very large encounter velocities we can use the so called, impulsive approximation. The impulsive approximation was introduced by L. Spitzer in 1958 to study the destruction of star clusters within the disk of our galaxy due to passing mass concentrations (this was before the discovery of giant molecular clouds). Lyman Spitzer ( ) 6

7 The Impulsive Approximation However, in the limit of very large encounter velocities we can use the so called, impulsive approximation. The basic idea is quite simple, we assume that the passing perturber moves so fast, that we can neglect the motion of the stars in the target. Despite the outrageous simplification that this may seem, this approximation is quite useful to make quick calculations that can guide us to see how large an effect we can expect. 7

8 The Impulsive Approximation However, in the limit of very large encounter velocities we can use the so called, impulsive approximation. M p We begin by centering our attention on the effect of the perturber on just one star within the target. p The relevant quantities are: M p, the mass of the perturber V col, the relative encounter velocity p, the impact parameter. v col 8

9 The Impulsive Approximation However, in the limit of very large encounter velocities we can use the so called, impulsive approximation. M p We begin by centering our attention on the effect of the perturber on just one star within the target. v col p R The relevant quantities are: M p, the mass of the perturber V col, the relative encounter velocity p, the impact parameter. We assume that the perturber zips by on a straight path from - to. By symmetry, it is only the component of the acceleration orthogonal to the path of the perturber that will affect the star. +$! v * = % a " dt = #$ % +$ GM p #$ R 2 p dt R 9

10 The Impulsive Approximation However, in the limit of very large encounter velocities we can use the so called, impulsive approximation. Simple geometry allows us to compute this integral: M p!! v * = ( 2GM ) p p! p 2 v col p R v col 10

11 The Impulsive Approximation However, in the limit of very large encounter velocities we can use the so called, impulsive approximation. Simple geometry allows us to compute this integral:!! v * = ( 2GM ) p p! p 2 v col We can split this equation as the product of two factors, whose physical units are acceleration and time, respectively. a Δt col! v * = GM p a max ( ) 2 p p 2 ( v col ) = a max! t col It is clear that the first factor is just the maximum acceleration, while the second factor can be interpreted as an interaction time, which is equal to the time it takes the perturber to advance twice the impact parameter. t 11

12 The Impulsive Approximation The result we obtained is for a point mass perturber, however, it is quite easy to consider an extended mass perturber:!! v * = ( 2GM ) p "(p)! p 2 v col p, "(p) # ' & µ p ( p$ ) d$ 1 $ 2 $ 2 %1 where χ(p) is a correction factor that depends on the impact parameter and µ p is the normalized, cumulative mass function of the perturber. 12

13 The Impulsive Approximation Once we have an expression for the change in velocity, we can compute the resulting change in energy for the star in the target.! E * =!K +!W =!(m * v * 2 /2) Notice that we have neglected the change in potential energy. This is a simplification we can do under the impulse approximation. The change in energy for the star is then,!! E * = m * v *!! ( ) + (1/2) m *!v * 2 where we have now neglected the linear term in the change in kinetic energy, because when we integrate over all stars in the target, this dot product will average to zero. v * 13

14 The Impulsive Approximation We can now substitute the expression we found for the change in velocity for the star to find the average change in kinetic energy <! E * > = 2( GM ) 2 p p v col r * p p g The last step is to integrate the effect over the entire target system by doing an integral along the path of the perturber +#! E tot = $! E * dz =!E tot (M p,v col, p g ) "# This integral can be obtained from simple geometrical arguments that relate the individual impact parameter for each star with that of the entire system. z 14

15 Tidal Stripping Tidal stripping is the mass loss produced when two systems collide. NGC 4038: NRAO/AUI and J. Hibbard 15

16 Tidal Stripping Tidal stripping is the mass loss produced when two systems collide. The orbital energy dissipated within the interacting systems heats them up and this leads to mass loss. A key parameter to understand this phenomenon is the ratio between the collisional interaction time and the local dynamical timescale within the systems. 16

17 Tidal Stripping Tidal stripping is the mass loss produced when two systems collide. Stars in the outskirts of the target have long dynamical timescales and so they move only a fraction of their orbit while the collision with the perturber takes place. 17

18 Tidal Stripping Tidal stripping is the mass loss produced when two systems collide. Stars in the outskirts of the target have long dynamical timescales and so they move only a fraction of their orbit while the collision with the perturber takes place. Stars in the inner parts of the target have short dynamical timescales and so they can complete several turns in their orbits while the collision takes place. 18

19 Tidal Stripping Tidal stripping is the mass loss produced when two systems collide. Stars in the outskirts of the target have long dynamical timescales and so they move only a fraction of their orbit while the collision with the perturber takes place. Stars in the inner parts of the target have short dynamical timescales and so they can complete several turns in their orbits while the collision takes place. This is an important difference, because then the fast star appears to the perturber, not as a particle, but as a continuous loop. 19

20 Tidal Stripping Tidal stripping is the mass loss produced when two systems collide. This means that the perturber can distort the orbit of the slow star when pumping energy into the star s motion, while it can only push around the orbit of the fast star. This is a case of adiabatic invariance. While for the slow star the encounter is impulsive, the same collision is adiabatic for the fast star. 20

21 Tidal Stripping The net result is that the perturber s orbital energy dumped in the outer envelope of the target, goes mainly in thermal motion. While the energy dumped in the central part of the target goes into moving it as a whole. As a result, the inner, most bound part of the target, is pushed as a whole by the encounter, while its envelope gets puffed up by the orbital energy it has absorbed in its thermal reservoir. The envelope is left behind and becomes unbound. This is tidal stripping. Although conceptually simple, This is a a complex phenomenon and we must use N-body simulations. 21

22 Putting everything together: A satellite spiraling in As an example of what we have seen, we will study the case of a satellite in a circular orbit within a larger host that gets dragged in and destroyed by dynamical friction and tidal truncation The basic ingredients are: A model for the satellite:! s (r) = 3M s 4 "r o 3 ( ) 1 + r2 ( ) #5 / 2, $ s (r) = # GM s r o 1 + r 2 ( 1+r ) 3 2 ( ) #1/ 2, M r (s) = M s r This is a model proposed by Plummer in Its advantage is that it is quite simple and it is a good approximation to globular clusters and some dwarf galaxies. Its total mass is M s and r o is the core radius, within which the density goes roughly constant. As we will see, this is an important characteristic. 22

23 Putting everything together: A satellite spiraling in As an example of what we have seen, we will study the case of a satellite in a circular orbit within a larger host that gets dragged in and destroyed by dynamical friction and tidal truncation The basic ingredients are: We also need a model for the host:!(r) = ( v 2 ) circ 1 4"G, R 2 2 #(R) = $v circ 2 [1$ Log(R /R t )], M R = (v circ This is a singular model with flat rotation curve. Its form is quite simple, but it has the unfortunate characteristic that its mass is unbound. So we introduce a cut-off radius R t, beyond which we set the density to zero and so the total mass is given by (v 2 circ /G)R t We will assume that the satellite moves within R t /G) R 23

24 Putting everything together: A satellite spiraling in As an example of what we have seen, we will study the case of a satellite in a circular orbit within a larger host that gets dragged in and destroyed by dynamical friction and tidal truncation The basic ingredients are: We will assume that the satellite is in a circular orbit of radius R. The Jacobi radius r t, is then, 3(r t /R) 3 = (M s / M R ) We can now substitute the mass models for the host and the satellite to find the size of the latter as a function of position within the host. In doing this, we will make the simplifying assumption that, as the satellite spirals in and its tidal radius shrinks, the satellite inside the tidal radius is unaffected and it is just the layers outside r t that are lost. 24

25 Putting everything together: A satellite spiraling in The tidal truncation formula then becomes, r ( t R ) 3 = 1 3 M r ( s ) (r t ) M R = 1 3 GM s 2 v circ R! # " x t 1+x t 2 3 $ &, % where x t (r t /r o ). From this relation we get, r ( t R ) = GM s 2 R ( ) 1/ 3 x # t 3v circ The term in the left hand side can be written as, Substituting this back in the tidal formula,! " 1+x t 2 r t R = r t / r o R / r o = x t R / r o, $ & % x t = R r o ( GM s ) 1/ 3 2 R 3v circ ( ) 1/ 3 x t! 1 + x 2 2 t = 1 2 GM s R 1+x r o t 3v 2 circ 25

26 Putting everything together: A satellite spiraling in Isolating x t and using its definition. We finally obtain the tidal radius of the satellite as a function of position within the host, r t = ( GM s R 2 ) 2 / 3! 2 ro 2 3v circ Notice that the tidal radius is defined for a positive radical only. To see when is that the tidal radius shrinks to zero, we write the condition for a null radical as a tidal condition, 3(r o /R t ) 3 = (M s / M Rt ) We recognize this result as the tidal condition for a point mass whose mass is equal to the initial satellite mass within the tidal field of a point mass whose mass is equal to the total host mass enclosed within the satellite s orbit. The terminal radial position is then reached when the tidal radius is equal to the core radius 26

27 Putting everything together: A satellite spiraling in This is a relative simple concept: A Plummer model dropped within a flat rotation curve potential well, will survive until its Roche lobe size, computed using its original mass, is equal to its core radius. What is behind the existence of this terminal radial position within the host? As we mentioned before, the tidal radius condition can be interpreted as a condition that relates densities: the satellite density and the mean host density within the satellite s orbit. If the satellite does not have a central density higher than the corresponding host value, it will be destroyed before reaching the host s center. 27

28 Putting everything together: A satellite spiraling in Terminal galactocentric distance As an example of what we have done, we plot the globular clusters in our Galaxy in a plot of mass versus core radius. Diagonal lines are iso-terminal galactocentric distances computed with the equation we derived. A cluster can only reach up to a depth within the Galaxy given by the value of the its terminal distance. 28

29 Putting everything together: A satellite spiraling in Terminal galactocentric distance It is clear that the vast majority of clusters are dense enough at their centers to survive all the way to the inside of our Galaxy s bulge, the exception being a group of three or maybe five clusters, of which the most extreme example is Palomar 4. This is an interesting feature: despite their vast difference in mass, globular clusters have central densities comparable to those of our Galaxy. The formation process (or processes) must somehow be able to produce systems of comparable central density over the huge range in mass spanned from globular clusters to our Galaxy. 29

30 Putting everything together: A satellite spiraling in With the formulae we have derived, we can plot the shrinking size of a globular cluster as it moves inward within our Galaxy. Two sets of curves are shown, for cluster masses of 10 6 and 10 5 solar masses. Within each set, three core radii are considered: 0.1, 1 and 10 parsecs. We can see that clusters steadily shrink as they fall until the tidal field erosion reaches their core radius, at which point they disappear very quickly. 30

31 Putting everything together: A satellite spiraling in Here we plot the mass of the cluster as a function of galactocentric position for all the cases considered before. The striking feature is that the mass of the clusters remains pretty much the same until it meets its catastrophic demise at the terminal radius. This is because clusters have little of their mass in their outer envelopes which are first eroded by the tidal field. This will allow us to make a very convenient simplification, we will assume that a satellite remains with constant mass when we model its spiraling in. 31

32 Putting everything together: A satellite spiraling in We now turn our attention to the effect of dynamical friction. The model we are using for the host, that of a singular isothermal sphere, has an isotropic velocity dispersion equal to:! = v circ / 2 Substituting this value in the dynamical friction formula we saw in the previous lesson, we get the dynamical friction force exerted in our satellite: F DF =! Log(" )GM s 2 [ ] $!0.428 Log(") (GM s 2 /R 2 ) R 2 erf (1)! 2e X # 32

33 Putting everything together: A satellite spiraling in This force produces a torque, that in the case of a circular orbit that we will assume for the satellite, produces a variation in angular momentum given by,! = F DF R = (dl s /dt) = d /dt (M s Rv circ ) Substituting now the form for the dynamical force we found and using the simplification that the satellite s mass remains constant, we obtain the following differential equation: R(dR /dt) =!0.428Log(") (GM s /v circ ) which can be easily integrated to give, R(t) = R i 2! Log(")(GM s /v circ ) t, where R i is the initial radial position of the satellite. 33

34 Putting everything together: A satellite spiraling in The decay time is given by: t DF = [1.168/Log(!)] (v circ R i 2 /GM s ) The plot shows the resulting decay times as a function of radial distance for satellites of three different masses. 34

35 The End 35

36 Some final words 36

37 Some final words Innocent light minded men, who think that Astronomy can be learnt by looking at the stars without knowledge of Mathematics will, in the next life, be birds. Plato (Timaeus) 37

38 Heaven is where the police are British, the cooks French, the mechanics German, the lovers Italian, and it is all organized by the Swiss. Hell is where the chefs are British, the mechanics French, The lovers Swiss, The police German, And it is all organized by the Italians. The International School has been organized by a Briton, a Greek and a Zapotec. It s been quite close to heaven.. and hell. I mean the best of both worlds. Thanks very much Dave, Manolis and Omar, for a wonderful experience, and to the students that had to suffer my lectures. 38