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Integrals of Motion Key Point #15 from Bachelor Course KeyKey12-10-07 see PointPoint http://www.strw.leidenuniv.nl/˜ #15#10 fromfrom franx/college/ BachelorBachelor mf-sts-07-c4-5 Course:Course:12-10-07 see http://www.strw.leidenuniv.nl/˜ IntegralsIntegrals ofof franx/college/ MotionMotion mf-sts-07-c4-6 4.2 Constants and Integrals of motion Integrals (BT 3.1 p 110-113) Much harder to define. E.g.: 1 2 Energy (all static potentials): E(⃗x,⃗v)= 2 v + Φ First, we define the 6 dimensional “phase space” coor- dinates (⃗x,⃗v). They are conveniently used to describe Lz (axisymmetric potentials) the motions of stars. Now we introduce: L⃗ (spherical potentials) • Integrals constrain geometry of orbits. • Constant of motion: a function C(⃗x,⃗v, t) which is constant along any orbit: examples: C(⃗x(t1),⃗v(t1),t1)=C(⃗x(t2),⃗v(t2),t2) • 1. Spherical potentials: E,L ,L ,L are integrals of motion, but also E,|L| C is a function of ⃗x, ⃗v, and time t. x y z and the direction of L⃗ (given by the unit vector ⃗n, Integral of motion which is defined by two independent numbers). ⃗n de- • : a function I(x, v) which is fines the plane in which ⃗x and ⃗v must lie. Define coor- constant along any orbit: dinate system with z axis along ⃗n I[⃗x(t1),⃗v(t1)] = I[⃗x(t2),⃗v(t2)] ⃗x =(x1,x2, 0) I is not a function of time ! Thus: integrals of motion are constants of motion, ⃗v =(v1,v2, 0) but constants of motion are not always integrals of motion! → ⃗x and ⃗v constrained to 4D region of the 6D phase space. In this 4 dimensional space, |L| and E are con- E.g.: for a circular orbit ψ = Ω t + ψo, so that C = served. This constrains the orbit to a 2 dimensional t − ψ/Ω. space. Hence the velocity is uniquely defined for a C is constant of motion, but not an integral as it de- given ⃗x pends on t. 2 2 vr = ± 2(E − Φ) − L /r Constants of motion ! 6 for any arbitrary orbit: vψ = ±L/r Initial position (⃗x0,⃗v0) at time t = t0.Canalways be calculated back from ⃗x,⃗v, t. 12-10-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-7 12-10-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-8 • 2. Integrals in 2 dimensional flattened potentials 4 4 2 2 Examples: 0 0 -2 -2 Circular potential V (x, y)=V (⃗r) -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 Two integrals: E,Lz. x0=0.5 y0=0 vx0=0 vy0=1 x0=1 y0=0 vx0=0 vy0=1 2 2 y 4 4 Flattened potential V (x, y)=ln(x + a + 1) Only “classic” integral of motion: E 2 2 0 0 -2 -2 Figures on the next page show the orbits that one gets by integrating the equations of motion. -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 Clearly the orbits are regular and do not fill equipoten- x0=1.1 y0=0 vx0=0 vy0=1 x0=1.2 y0=0 vx0=0 vy0=1 tial surface the first 3 are box orbits (no net angular momen- tum) Furthermore, they do not traverse each point in a ran- dom direction, but generally only in 2 directions avoid outer x-axis Conclusion: the orbits do not occupy a 3 dimensional space in the 4-dimensional phase-space, but they 4 4 occopy only a 2-dimensional space ! 2 2 This indicates that there is an additional integral of 0 0 motion: ’a non-classical integral’ -2 -2 The non-classical integral, plus the regular ’Energy’, -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 constrain the orbit to lie on a 2 dimensional surface x0=2 y0=0 vx0=0 vy0=1 x0=4 y0=0 vx0=0 vy0=1 in the 4 dimensional phase-space. loop orbits (with net angular momentum) avoid inner x-axis can circulate in two directions. 12-10-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-9 12-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-10 A homogeneous ellipsoid Orbits for general homogeneous ellipsoid All orbits are box orbits The homogeneous ellipsoid helps us to understand how additional integrals of motions, and box orbits, exist. 4 4 Consider a density distribution: 2 2 0 0 2 ρ = ρ0H(1 − m ), with -2 -2 2 x2 y2 z2 -4 -4 m = a2 + b2 + c2 -4 -2 0 2 4 -4 -2 0 2 4 x0=0.5 y0=0 vx0=0 vy0=1 x0=1 y0=0 vx0=0 vy0=1 H(x)=1for x ≥ 0, H(x)=0for x<0 4 4 2 2 Potential inside the ellipsoid: 0 0 2 2 2 -2 -2 V = Axx + Ayy + Azz + C0 -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 x0=1.1 y0=0 vx0=0 vy0=1 x0=1.2 y0=0 vx0=0 vy0=1 Forces are of the form Fi = −Aixi,i.e.3inde- pendent harmonic oscillators: 4 4 2 2 0 0 xi = ai cos(ωit + ψ0,i) -2 -2 -4 -4 -4 -2 0 2 4 -4 -2 0 2 4 3 integrals of motion, Ei x0=2 y0=0 vx0=0 vy0=1 x0=4 y0=0 vx0=0 vy0=1 Hence the orbits are as shown on the next page 12-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-11 12-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-12 4 • 3. Axisymmetric potentials (BT 3.2) Φ = Φ(R, |z|), with R2 = x2 + y2. 2 For z=0: orbits as if potential were circular For (R,z): 3D orbit can be described as 2D orbit in 0 meridional plane: d2⃗r -2 = −∇Φ(R, z) dt2 -4 with ⃗er,⃗eψ,⃗ez unit vectors in r, ψ and z direction: -4 -2 0 2 4 x0=0.5x0=1.1x0=1.2x0=1x0=2x0=4 y0=0y0=0 vx0=0vx0=0 vy0=1vy0=1 ⃗r = R ⃗e + z ⃗e Special case: a=b r z all orbits are loop orbits δΦ δΦ ∇Φ = ⃗e + ⃗e δR r δz z The equations of motion reduce to d2R −δΦ − Rψ˙ 2 = dt2 δR d δΦ R2ψ˙ = − =0 dt δψ d2z δΦ = − dt2 δz From the second equation we see that Lz is constant. ˙ 2 2 Hence ψ = Lz/R . If we use this in the upper equa- tion we obtain for the motion in two dimensions R, z: 12-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-13 12-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-14 d2R δΦ d2z δΦ = − eff = − eff dt2 δR dt2 δz 2 Lz with Φeff = Φ(R, z)+ 2R2 . Hence 3D motion can be reduced to motion in (R,z) plane or meridional plane, under influence of the effective potential Φeff . Total energy: 2 2 E = Φeff +1/2R˙ +1/2˙z . Allowed region in meridional plane For energy E motion only allowed in area where Φeff <E. E.g. 1 2 2 z2 If Φ = 2 v0 ln(R + q2 ), then 2 2 1 2 2 z Lz Φ = v0 ln(R + )+ eff 2 q2 2R2 , i.e. only motion if E>Φeff . 12-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-15 12-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-16 Lines of constant Φeff are shown in Figure 3.2. Stars integral of motion must play a role → dimensionality with energy E have zero velocity at curves of Φeff = reduced to 3 (e.g. R, z, ψ). E. This integral is a non-classical integral of motion Homework Assignments: 1) Box orbits are characterized by the fact they they go through the center, and have no net angular momen- tum. Explain how it comes that they don’t have net angular momentum, even though a star in a box orbit has a non-zero angular momentum at most times. 2) How is it possible that the box orbit touches the equipotential surface given by Energy = Φ? 3) Why does a loop orbit not touch that surface ? 4) Why are there no loop orbits in a homogeneous el- lipsoid ? 5) Calculate at least 3 orbits in the 2-dimentional po- tential Φ =ln(1+x2 + y2/2). Do this as follows: Start the star at a given location (x0, 0) with velocity (0,v0). Calculate the shift in position at time dt by dt*⃗v.Cal- culate the shift in velocity at time dt by dt*F⃗ .And keep integrating ! Again: not all orbits fill the space Φeff <Efully! Two integrals (E,Lz) reduce the dimensionality of the orbit from 6 to 4 (e.g. R, z, ψ, vz). Therefore another 12-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-17 12-10-07see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c4-18 A general 3-dimensional potential A Simple recipe to build galaxies Schwarzschild’s method: 2 2 • Define density ρ St¨ackel potential( ρ =1/(1 + m ) ) • Calculate potential, forces • Integrate orbits, find orbital densities ρi • Calculate weights wi > 0 such that ρ = ρiwi ! Examples: build a 2D galaxy in a logarithmic potential Φ =ln(1+x2 + y2/a).
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