Binary Objects in the Kuiper Belt and Outlying Centaurs: Simulations

Binary Objects in the Kuiper Belt and Outlying Centaurs: Simulations Stephan Kolassa [email protected] Institut f¨ ur Angewandte Mathematik Friedrich-Schiller-Universit¨at Jena Germany September 16, 2004

Abstract. Two exchange reaction scenarios to account for the characteristics of Kuiper belt binaries and the orbital data of the centaurs 2000 CR 105 and 2003 VB 12 (Sedna) have recently been proposed. The results from the literature are replicated, and new simulations are reported.

Contents 1 Introduction and Fundamentals 1.1 Introduction . . . . . . . . . . . . . . . . . 1.2 Mathematical and Physical Fundamentals 1.3 Numerics . . . . . . . . . . . . . . . . . . 1.4 Comparison of Different Integrators . . . 1.5 Literature, Websites and Programs . . . . 2 Binary Objects in the Kuiper Belt 2.1 The Kuiper Belt . . . . . . . 2.2 Two Bodies . . . . . . . . . . 2.3 Simulations and Results . . . 2.4 Possibilities for Further Work

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3 Outlying Centaurs: Capture of Extrasolar Planetesimals 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . 3.3 A Slow Pass . . . . . . . . . . . . . . . . . . . . . . . 3.4 A Medium-Speed Pass . . . . . . . . . . . . . . . . . 3.5 A Fast Pass . . . . . . . . . . . . . . . . . . . . . . . 3.6 Discussion and Possibilities for Further Work . . . .

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59

1

1 Introduction and Fundamentals 1.1 Introduction Our knowledge about the Solar System has expanded greatly in the past decades. Belts and clouds of planetesimals postulated by theoreticians 50 years ago have been observed and subjected to intense study. However, new observations increasingly call accepted models of orbit formation into question. In this paper, we deal with two topics. On the one hand, we consider a recently proposed model to account for the orbital characteristics of binary objects in the Kuiper belt, between 30 and 50 AU from the sun. On the other hand, a different model has been offered to explain the orbit formation of two centaurs. We replicate the findings of the models’ authors and give some further simulation results. A word of caution is in order: the author is no (astro-)physicist or astronomer and has no special experience with N -body systems. Our approach is rather more “interesting phenomena”-driven than a rigorous critique of the models we consider. Nevertheless, we hope that this paper shows that even non-specialists can understand the questions astronomers work on and that the layman is able to conduct his own dynamical experiments.

1.2 Mathematical and Physical Fundamentals 1.2.1 The Differential Equations Consider N point bodies with masses m1 , . . . , mN . At time t, the bodies are at the positions x1 = x1 (t), . . . , xN = xN (t) ∈ R3 . We call the first derivative of the positions with respect to time, x˙ i , the velocity of the ith body at time t, the second derivative x ¨i ... its acceleration and the third derivative x i its jerk. By Newton’s First Law of Motion, the force F acting on a body with mass m causes an acceleration x ¨ as follows: F = m¨ x. By the Law of Gravity, the j th body exerts the following gravitational force on the ith body: Fji = Gmi mj where G = 6.67 · 10−11

m3 kg·s2

xj − xi , ||xj − xi ||3

is the universal gravitational constant.

Thus, our differential equation system is as follows: mi x ¨i =

X j6=i

Gmj mi

xj − xi ||xj − xi ||3

2

for 1 ≤ i ≤ N .

For our purely qualitative analyses, we set G = 1. With free initial conditions, our system now becomes: X xj − xi mj x ¨i = with xi (0) = x0i , x˙ i (0) = x˙ 0i for 1 ≤ i ≤ N . (1) ||xj − xi ||3 j6=i

By the Theorem on Existence and Uniqueness of Ordinary Differential Equations, this system has a unique solution. However, it will usually not be possible to give a closed form for this solution. Further remarks are given by Guthmann (2000). 1.2.2 The Semi-Major Axis and Eccentricity Since we are mostly interested in newly formed binaries and since the binaries found in the Kuiper belt are characterized by their large eccentricities, we will now show how we can derive the semi-major axis and the eccentricity of a binary without having to transform its trajectory into a plane ellipse. This discussion is adapted from Hut and Makino (2003a), section 11.1. Suppose that a binary is composed of the two masses mi and mj . Let r := xj −xi denote the relative position vector pointing from body i to j and v := x˙ j − x˙ i the relative velocity vector, i.e., v = r. ˙ Now, the total energy of a binary system is the sum of its kinetic and its potential energy. Adopting the convention that an object “at infinity” has zero potential energy, we obtain the following formula for the total energy of a two-body system: E = Ekin + Epot =

Gmi mj 1 mi mj ||v||2 − . 2 mi + mj ||r||

(2)

Next, the angular momentum of the binary system is as follows: mi mj L= r × v. mi + mj We note that we could also have defined r and v as pointing from object j to object i without changing the formulae for the energy and the angular momentum. We simplify the above equations by introducing the reduced mass of the binary: mi mj µ := . mi + mj Then the energy and the angular momentum per unit of reduced mass become e := E = 1 ||v||2 − G mi + mj E µ 2 ||r||

e := L = r × v. and L µ

e in terms of the semi-major axis a and L e in terms of a Now, it is possible to express E and the eccentricity e: e = −G mi + mj E 2a

e 2 = G(mi + mj )a(1 − e2 ). and ||L||

3

Solving for a and e, we obtain mi + mj a = −G e 2E

s and e =

1−

e 2 ||L|| . G(mi + mj )a

1.3 Numerics There are three well accepted basic schemes for integrating ordinary differential equations (Press et al., 1992): I Runge-Kutta methods I Richardson extrapolation and its particular implementation as the BulirschStoer method I predictor-corrector methods. A brief description of each of these types follows. 1. Runge-Kutta methods propagate a solution over an interval by combining the information from several Euler-type steps (each involving one evaluation of the right-hand f ’s), and then using the information obtained to match a Taylor series expansion up to some higher order. 2. Richardson extrapolation uses the powerful idea of extrapolating a computed result to the value that would have been obtained if the stepsize had been very much smaller than it actually was. In particular, extrapolation to zero stepsize is the desired goal. The first practical ODE integrator that implemented this idea was developed by Bulirsch and Stoer, and so extrapolation methods are often called Bulirsch-Stoer methods. 3. Predictor-corrector methods store the solution along the way, and use those results to extrapolate the solution one step advanced; they then correct the extrapolation using derivative information at the new point. These are best for very smooth functions. Press et al. go on to explain that Bulirsch-Stoer methods have been replacing predictorcorrector in many applications, and that only rather sophisticated predictor-corrector routines appear competitive. Accordingly, they do not give an implementation of a predictor-corrector scheme. However, Schlitt (2000) compares the different integrators in their specific application to N -body problems and comes to the conclusion that even simple predictor-corrector methods may be appropriate in this case for the following reasons: I The right-hand-side in (1) is expensive to evaluate, and predictor-corrector schemes typically need fewer evaluations than other methods. This frees up computation time, which can be spent on precision.

4

I Expensive methods require large step sizes to keep computation time low, and at large step sizes, close encounters of bodies look like singularities to the integrator. However, since the bodies will move around each other, small step sizes will show that all functions are smooth, thus fulfilling the key requirement for predictorcorrector methods to be competitive. 1.3.1 nbody sh1 We used the program nbody sh1 by Hut and Makino (2002), which is documented line by line in great detail by Hut and Makino (2003b). Below, we will present the algorithm used to perform the next integration step at time t. nbody sh1 uses adaptive step size control. It is called with a step size control parameter dcontrol via the -d option. By default, dcontrol = 0.03, and one should use dcontrol  1. At time t, nbody sh1 has an estimate of the time increment until “something drastic happens”, such as a collision of two particles. This time increment is called θcoll and is determined in the course of the step at time t, see below. First, the step size θ at time t is determined as θ := dcontrol · θcoll . Now the predictor step is performed to predict the positions and velocities at time t + θ. This is nothing but a simple Taylor expansion: 1 1 ... xpred (t + θ) := xi (t) + x˙ i (t)θ + x ¨i (t)θ2 + x i (t)θ3 i 2 6 1 ... pred 2 x˙ i (t + θ) := x˙ i (t) + x ¨i (t)θ + x i (t)θ 2

for 1 ≤ i ≤ N for 1 ≤ i ≤ N .

Next, the accelerations and jerks are updated based on the predicted positions and velocities: pred rji := xpred (t + θ) − xpred (t + θ) j i pred vji := x˙ pred (t + θ) − x˙ pred (t + θ) j i

x ¨pred (t i

+ θ) :=

pred X mj rji j6=i

X ...pred x i (t + θ) := j6=i

for 1 ≤ i ≤ N

pred 3 ||rji ||

mj pred 3 ||rji ||

 ·

pred vji

−

pred pred pred 3hrji , vji irji pred 2 ||rji ||

 for 1 ≤ i ≤ N .

(3)

The acceleration value results directly from Newton’s Law, while the jerk formula is a simple differentiation of the acceleration. Now the collision time variable is reset: s   pred pred 3 ||rji || ||rji || 1≤i

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