Stellar Dynamics and Structure of Galaxies

Transcription

1 Stellar Dynamics and Structure of Galaxies in a given potential Vasily Belokurov vasily@ast.cam.ac.uk Institute of Astronomy Lent Term / 59

2 1 Collisions Model requirements 2 in spherical 3 4 Orbital periods Example Outline I 2 / 59

3 Galaxies Part II Collisions Do we have to worry about collisions? Collisions Model requirements Globular clusters look densest, so obtain a rough estimate of collision timescale for them 3 / 59

4 Collisions Model requirements Collisions in globular clusters The case of NGC 2808 ρ M pc 3 M 0.8 M. n pc 3 is the star number density. We have σ r 13 km s 1 as the typical 1D speed of a star, so the 3D speed is 3 σ r (= σx 2 + σy 2 + σz 2 ) 20 km s 1. Since M R (see Fluids, or Stars, course notes), have R 0.8R. 4 / 59

5 Collisions Model requirements Collisions in globular clusters The case of NGC 2808 For a collision, need the volume π(2r ) 2 σt coll to contain one star, i.e. n 0 = 1/ ( π(2r ) 2 σt coll ) or (1.1) t coll = 1/ ( 4πR σn 2 ) 0 (1.2) R = 0.8R, n 0 = 10 5 pc 3, σ = 20km/s Putting in the numbers gives t coll yr. So direct collisions between stars are rare, but if you have 10 6 stars then there is a collision every 10 8 years, so they do happen. Note that NGC 2808 is 10 times denser than typical So, for now, ignore collisions, and we are left with stars orbiting in the potential from all the other stars in the system. 5 / 59

6 Collisions Model requirements Model requirements Model (e.g., a globular cluster) just as a self-gravitating collection of objects. Have a gravitational potential well Φ(r), approximately smooth if the number of particles >> 1. Conventionally take Φ( ) = 0. Stars orbit in the potential well, with time per orbit (for a globular cluster) 2R h /σ 10 6 years << age. Remember how to measure age for globular clusters? Stars give rise to Φ(r) by their mass, so for this potential in a steady state could average each star over its orbit to get ρ(r). The key problem is therefore self-consistently building a model which fills in the terms: Φ(r) stellar orbits ρ(r) Φ(r) (1.3) Note that in most observed cases we only have v line of sight (R), so it is even harder to model real systems. Self-consistent = orbits & stellar mass give ρ, which leads to Φ, which supports the orbits used to construct ρ 6 / 59

7 in spherical The law of attraction s of motion and Newtonian gravity GR not needed, since 10 < v < 10 3 km is << c = 3 10 s 5 km s GM rc 2??? The gravitational force per unit mass acting on a body due to a mass M at the origin is f = GM ˆr = GM r 2 r 3 r (1.4) We can write this in terms of a potential Φ, using ( ) 1 = 1 r r 2ˆr (1.5) 7 / 59

8 in spherical The corresponding potential So f = Φ (1.6) where Φ is a scalar, Φ = Φ(r) = GM (1.7) r Hence the potential due to a point mass M at r = r 1 is Φ(r) = GM r r 1 (1.8) 8 / 59

9 Density vs Potential in spherical From Hayashi et al, The shape of the gravitational potential in cold dark matter haloes 9 / 59

10 Galaxies Part II The law of motion in spherical 10 / 59

11 in spherical Particle of constant mass m at position r subject to a force F. : i.e. If F is due to a gravitational potential Φ(r), then d (mṙ) = F (1.9) dt m r = F (1.10) F = mf = m Φ (1.11) The angular momentum about the origin is H = r (mṙ).then dh dt where G is the torque about the origin. = r (m r) + mṙ ṙ = r F G (1.12) 11 / 59

12 in spherical Particle of constant mass m at position r subject to a force F. : i.e. If F is due to a gravitational potential Φ(r), then d (mṙ) = F (1.9) dt m r = F (1.10) F = mf = m Φ (1.11) The angular momentum about the origin is H = r (mṙ).then dh dt where G is the torque about the origin. = r (m r) + mṙ ṙ = r F G (1.12) 11 / 59

13 in spherical The kinetic energy Energy T = 1 2 mṙ. ṙ (1.13) dt dt = mṙ. r = F. ṙ (1.14) If F = m Φ, then dt dt = mṙ. Φ(r) (1.15) But if Φ is independent of t, the rate of change of Φ along an orbit is d dt Φ(r) = Φ. ṙ (1.16) from the chain rule 12 / 59

14 in spherical The kinetic energy Energy T = 1 2 mṙ. ṙ (1.13) dt dt = mṙ. r = F. ṙ (1.14) If F = m Φ, then dt dt = mṙ. Φ(r) (1.15) But if Φ is independent of t, the rate of change of Φ along an orbit is d dt Φ(r) = Φ. ṙ (1.16) from the chain rule 12 / 59

15 in spherical Hence Energy dt dt = m d Φ(r) (1.17) dt m d ( ) 1. ṙ + Φ(r) = 0 (1.18) dt 2ṙ E = 1 2ṙ. ṙ + Φ(r) (1.19) The total energy is constant for a given orbit 13 / 59

16 in spherical Hence Energy dt dt = m d Φ(r) (1.17) dt m d ( ) 1. ṙ + Φ(r) = 0 (1.18) dt 2ṙ E = 1 2ṙ. ṙ + Φ(r) (1.19) The total energy is constant for a given orbit 13 / 59

17 in spherical Φ(r) = Φ( r ) = Φ(r), so f = Φ = ˆr dφ dr. The orbital angular momentum H = mr ṙ, and dh dt in spherical dφ = r mf = m r ˆr = 0. (1.20) dr So the angular momentum per unit mass h = H/m = r ṙ is a constant vector, and is perpendicular to r and ṙ the particle stays in a plane through the origin which is perpendicular to h Check: r h, r + δr = r + ṙδt h since both r and ṙ h, so particle remains in the plane Thus the problem becomes a two-dimensional one to calculate the orbit use 2-D cylindrical coordinates (R, φ, z) at z = 0, or spherical polars (r, θ, φ) with θ = π 2. So, in 2D, use (R, φ) and (r, φ) interchangeably.. 14 / 59

18 in spherical The equation of motion in can be written in radial angular terms, using r = rˆr = rê r + 0ê φ, so r =(r, 0). We know that d dt êr = φê φ (1.21) and d dt êφ = φê r (1.22) ê r = cos(φ)ê x + sin(φ)ê y ê φ = sin(φ)ê x + cos(φ)ê y d dt êr = sin(φ) φê x + cos(φ) φê y d dt êφ = cos(φ) φê x sin(φ) φê y 15 / 59

19 in spherical The equation of motion in can be written in radial angular terms, using r = rˆr = rê r + 0ê φ, so r =(r, 0). We know that d dt êr = φê φ (1.21) and d dt êφ = φê r (1.22) ê r = cos(φ)ê x + sin(φ)ê y ê φ = sin(φ)ê x + cos(φ)ê y d dt êr = sin(φ) φê x + cos(φ) φê y d dt êφ = cos(φ) φê x sin(φ) φê y 15 / 59

20 in spherical Hence and so ṙ = ṙê r + r φê φ (1.23) or ṙ = v =(ṙ, r φ) r = rê r + ṙ φê φ + ṙ φê φ + r φê φ r φ 2 ê r = ( r r φ 2 )ê r + 1 d ( ) r 2 φ ê φ r dt = a = [ r r φ 2, 1 d ( ) r 2 φ ] r dt (1.24) In general f =(f r, f φ ), and then f r = r r φ 2, where the second term ( ) is the centrifugal force, since we are in a rotating frame, and the torque rf φ = d dt r 2 φ (= r f). In a spherical potential f φ = 0, so r 2 φ is constant. 16 / 59

21 in spherical To determine the shape of the orbit we need to remove t from the equations and find r(φ). It is simplest to set u = 1/r, and then from r 2 φ = h obtain Then and φ = hu 2 (1.25) ṙ = 1 u 2 u = 1 u 2 du dφ φ = h du dφ (1.26) r = h d 2 u dφ 2 φ = h 2 u 2 d 2 u dφ 2. (1.27) 17 / 59

22 in spherical So the radial equation of motion becomes r r φ 2 = f r h 2 u 2 d 2 u dφ 2 1 u h2 u 4 = f r (1.28) d 2 u dφ 2 + u = f r h 2 u 2 (1.29) The orbit equation in spherical potential 18 / 59

23 in spherical Since f r is just a function of r (or u) this is an equation for u(φ), i.e. r(φ) - the path of the orbit. Note that it does not give r(t), or φ(t) - you need one of the other equations for those. If we take f r = GM r = GMu 2, then 2 d 2 u dφ 2 + u = GM/h2 (1.30) (which is something you will have seen in the Relativity course). 19 / 59

24 in spherical The solution to this equation is Kepler orbits Solution to the equation of motion l r = lu = 1 + e cos(φ φ 0) (1.31) which you can verify simply by putting it in the differential equation. Then e cos(φ φ 0) l e cos(φ φ 0) l = GM h 2 so l = h 2 /GM and e and φ 0 are constants of integration. 20 / 59

25 in spherical 1 r = 1 + e cos(φ φ 0) l Kepler orbits Bound orbits l Note that if e < 1 then 1/r is never zero, so r is bounded in the range 1+e < r < l 1 e. Also, in all cases the orbit is symmetric about φ = φ 0, so we take φ 0 = 0 as defining the reference line for the angle φ. l is the distance from the origin for φ = ± π 2 (with φ measured relative to φ 0 ). 21 / 59

26 in spherical Kepler orbits Bound orbits We can use different parameters. Knowing that the point of closest approach (perihelion for a planet in orbit around the Sun, periastron for something about a star) is at l/(1 + e) when φ = 0 and the aphelion (or whatever) is at l/(1 e) when φ = π, we can set the distance between these two points (= major axis of the orbit)=2a. Then l 1 + e + l 1 e = 2a l(1 e) + l(1 + e) = 2a(1 e2 ) (1.32) l = a(1 e 2 ) (1.33) r P = a(1 e) is the perihelion distance from the gravitating mass at the origin, and r a = a(1 + e) is the aphelion distance. The distance of the Sun from the midpoint is ae, and the angular momentum h 2 = GMl = GMa(1 e 2 ). 22 / 59

27 in spherical The energy per unit mass E = 1. ṙ + Φ(r) = ṙ 2ṙ 2 r 2 φ 2 GM (1.34) r r p = a(1 e) This is constant along the orbit, so we can evaluate it anywhere convenient - e.g. at perihelion where ṙ = 0. Then φ = h and so rp 2 E = 1 GMa(1 e 2 ) 2 a 2 (1 e) 2 GM a(1 e) = GM [ ( ) e 1 ] a 2 1 e 1 e = GM 2a (1.35) This is < 0 for a bound orbit, and depends only on the semi-major axis a (and not e). 23 / 59

28 in spherical... deduced from observations, and explained by Newtonian theory of gravity. 24 / 59

29 in spherical 1 are ellipses with the Sun at a focus. 2 Planets sweep out equal areas in equal time δa = 1 2 r 2 δφ [= 1 r(rδφ)] (1.36) 2 da dt = 1 2 r 2 h φ = = constant (1.37) 2 Kepler s second law is a consequence of a central force, since this is why h is a constant. 25 / 59

30 in spherical 1 are ellipses with the Sun at a focus. 2 Planets sweep out equal areas in equal time δa = 1 2 r 2 δφ [= 1 r(rδφ)] (1.36) 2 da dt = 1 2 r 2 h φ = = constant (1.37) 2 Kepler s second law is a consequence of a central force, since this is why h is a constant. 25 / 59

31 in spherical 3 (Period) 2 (size of orbit) 3 ( In one period T, the area swept out is A = 1 2 ht = T 0 But A = area of ellipse = πab = πa 2 1 e 2 [ Have π 0 = l2 2 A = = 2π 0 2π 0 2π 0 r dφ r 2 dφ rdr dφ (1 + e cos φ) 2 ) da dt dt 3rd Law l r = lu = 1 + e cos(φ φ 0) dx (a + b cos x) 2 = π a a 2 b 2 a2 b 2 26 / 59

32 in spherical so Since l = a(1 e 2 ) this implies and since b = a 1 e 2, ] A = 2 l2 2 π 1 e e 2 A = πa 2 1 e 2 A = πab 3rd Law def: e = 1 b2 a 2 27 / 59

33 in spherical Therefore T = 2A h = 2πa2 1 e 2 h 2πa 2 1 e = 2 GMa(1 e2 ) 3rd Law since h 2 = GMa(1 e 2 ) a 3 T = 2π GM T 2 a 3 (1.38) where in this case M is the mass of the Sun. Note: Since E = GM 2πGM 2a, the period T =. ( 2E) / 59

34 in spherical What happens to l r = 1 + e cos φ when e 1? If e > 1 then 1 + e cos φ = 0 has solutions φ where r = cos φ = 1/e Then φ φ φ, and, since cos φ is negative, π 2 < φ < π. The orbit is a hyperbola. If e = 1 then the particle just gets to infinity at φ = ±π - it is a parabola. 29 / 59

35 in spherical What happens to l r = 1 + e cos φ when e 1? If e > 1 then 1 + e cos φ = 0 has solutions φ where r = cos φ = 1/e Then φ φ φ, and, since cos φ is negative, π 2 < φ < π. The orbit is a hyperbola. If e = 1 then the particle just gets to infinity at φ = ±π - it is a parabola. 29 / 59

36 in spherical What happens to l r = 1 + e cos φ when e 1? If e > 1 then 1 + e cos φ = 0 has solutions φ where r = cos φ = 1/e Then φ φ φ, and, since cos φ is negative, π 2 < φ < π. The orbit is a hyperbola. If e = 1 then the particle just gets to infinity at φ = ±π - it is a parabola. 29 / 59

37 Kepler orbits in spherical 30 / 59

38 in spherical Energies for these unbound orbits: So, as r E 1 2ṙ 2 E = h 2 2ṙ 2 r 2 GM r r 2 φ = h 31 / 59

39 in spherical Recall d dt of this and since h = r 2 φ l r = 1 + e cos φ l ṙ = e sin φ φ r 2 ṙ = eh l sin φ As r cos φ 1/e E 2ṙ2 1 = 1 e 2 h 2 (1 2 l 2 1e ) 2 = GM 2l (e2 1) (recalling that h 2 = GMl) Thus E > 0 if e > 1 and for parabolic orbits (e = 1) E = / 59

40 in spherical We have seen that in a fixed potential Φ(r) a particle has constant energy E = 1 2ṙ2 + Φ(r) along an orbit. If we adopt the usual convention and take Φ(r) 0 as r, then if at some point r 0 the particle has velocity v 0 such that 1 2 v2 0 + Φ(r 0 ) > 0 then it is able to reach infinity. So at each point r 0 we can define an escape velocity v esc such that v esc = 2Φ(r 0 ) 33 / 59

41 in spherical The escape velocity from the Sun ( ) 1 2GM 2 ( r0 ) 1 2 v esc = = 42.2 a.u. Note: The circular velocity v circ is such that r φ 2 = GM r 2 r φ GM ( r0 ) 1 2 = v circ = = 29.8 r 0 a.u. r 0 From the Solar neighborhood km s 1 km s 1 (= 2π a.u./yr). v esc = 2v circ for a point mass source of the gravitational potential. 34 / 59

42 Galaxies Part II From the Galaxy in spherical 35 / 59

43 Kepler orbits in spherical 36 / 59

44 What we have done so far is assume a potential due to a fixed point mass which we take as being at the origin of our polar coordinates. We now wish to consider a situation in which we have two point masses, M 1 and M 2 both moving under the gravitational attraction of the other. This is a cluster of N stars where N = 2 and we can solve it exactly! Hooray! The potential is no longer fixed at origin Φ(r) = GM 1 r r 1 GM 2 r r 2 37 / 59

45 What we have done so far is assume a potential due to a fixed point mass which we take as being at the origin of our polar coordinates. We now wish to consider a situation in which we have two point masses, M 1 and M 2 both moving under the gravitational attraction of the other. This is a cluster of N stars where N = 2 and we can solve it exactly! Hooray! The potential is no longer fixed at origin Φ(r) = GM 1 r r 1 GM 2 r r 2 37 / 59

46 What we have done so far is assume a potential due to a fixed point mass which we take as being at the origin of our polar coordinates. We now wish to consider a situation in which we have two point masses, M 1 and M 2 both moving under the gravitational attraction of the other. This is a cluster of N stars where N = 2 and we can solve it exactly! Hooray! The potential is no longer fixed at origin Φ(r) = GM 1 r r 1 GM 2 r r 2 37 / 59

47 Or the force acting on star 1, due to star 2 is F 1 = GM 1M 2 r 1 r 2 2 in the direction of r 2 r 1 And by symmetry (or Newton s 3rd law) F 1 = GM 1M 2 r 1 r 2 3 (r 2 r 1 ) F 2 = GM 1M 2 r 1 r 2 3 (r 1 r 2 ) 38 / 59

48 Then we know and where M 1 r 1 = GM 1M 2 d 2 ˆd (1.39) M 2 r 2 = GM ( ) 1M 2 d 2 ˆd is the vector from M 2 to M 1. Using these two we can write for d = r 1 r 2 (1.40) d = r 1 r 2 (1.41) d = G(M 1 + M 2 ) d 2 ˆd (1.42) 39 / 59

49 d = G(M 1 + M 2 ) ˆd which is identical to the equation of motion of a particle subject to a fixed mass M 1 + M 2 at the origin. So we know that the period a T = 2π 3 (1.43) G(M 1 + M 2 ) d 2 where the size (maximum separation) of the relative orbit is 2a. 40 / 59

50 d = G(M 1 + M 2 ) ˆd which is identical to the equation of motion of a particle subject to a fixed mass M 1 + M 2 at the origin. So we know that the period a T = 2π 3 (1.43) G(M 1 + M 2 ) d 2 where the size (maximum separation) of the relative orbit is 2a. 40 / 59

51 If we take the coordinates for the centre of mass r CM = From equations (1.39) and (1.40) we know that M 1 M 2 r 1 + r 2 (1.44) M 1 + M 2 M 1 + M 2 and so or i.e.ṙ CM =constant. M 1 r 1 + M 2 r 2 = 0 (1.45) d dt (M 1ṙ 1 + M 2 ṙ 2 ) = 0 (1.46) (M 1 ṙ 1 + M 2 ṙ 2 ) = constant (1.47) We can choose an inertial frame in which the centre of mass has zero velocity, so might as well do so 41 / 59

52 If we take the coordinates for the centre of mass r CM = From equations (1.39) and (1.40) we know that M 1 M 2 r 1 + r 2 (1.44) M 1 + M 2 M 1 + M 2 and so or i.e.ṙ CM =constant. M 1 r 1 + M 2 r 2 = 0 (1.45) d dt (M 1ṙ 1 + M 2 ṙ 2 ) = 0 (1.46) (M 1 ṙ 1 + M 2 ṙ 2 ) = constant (1.47) We can choose an inertial frame in which the centre of mass has zero velocity, so might as well do so 41 / 59

53 Note that choosing r CM = 0 M 1 r 1 = M 2 r 2, and so r 1 = d + r 2 = d M1 This r 1 = M2 M 1+M 2 d and similarly r 2 = M1 M 1+M 2 d. The angular momentum J (or H if you want) is M 2 r 1 J = M 1 r 1 ṙ 1 + M 2 r 2 ṙ 2 = = M 1 M2 2 (M 1 + M 2 ) 2 d ḋ + M 2M1 2 (M 1 + M 2 ) 2 d ḋ M 1 M 2 d M 1 + M ḋ 2 (1.48) So J = µh (1.49) where µ is the reduced mass, and h is the specific angular momentum. 42 / 59

54 Note that choosing r CM = 0 M 1 r 1 = M 2 r 2, and so r 1 = d + r 2 = d M1 This r 1 = M2 M 1+M 2 d and similarly r 2 = M1 M 1+M 2 d. The angular momentum J (or H if you want) is M 2 r 1 J = M 1 r 1 ṙ 1 + M 2 r 2 ṙ 2 = = M 1 M2 2 (M 1 + M 2 ) 2 d ḋ + M 2M1 2 (M 1 + M 2 ) 2 d ḋ M 1 M 2 d M 1 + M ḋ 2 (1.48) So J = µh (1.49) where µ is the reduced mass, and h is the specific angular momentum. 42 / 59

55 Momentum loss due to mass loss 43 / 59

56 Momentum loss due to Gravitational Radiation 44 / 59

57 Momentum loss due to Gravitational Radiation Question: predict the evolution of the pulsar s orbit. 45 / 59

58 Momentum loss due to Gravitational Radiation Weisberg and Taylor / 59

59 Binary Super-massive Black holes 47 / 59

60 Orbital periods Example Remember the orbit equation? ( d 2 u 1 ) dφ 2 + u = f u h 2 u 2 (1.50) where u 1 r and f r = f for a spherical potential. For f from a gravitational potential, we have ( ) 1 f = dφ = u 2 dφ u dr du (1.51) since gravity is conservative. There are two types of orbit: Unbound: r, u 0 as φ φ Bound: r (and u) oscillate between finite limits. 48 / 59

61 Energy Orbital periods Example If we take (1.50) du dφ : du d 2 u dφ dφ 2 + u du dφ + u2 dφ du h 2 u 2 du dφ = 0 (1.52) [ d ( ) ] 2 1 du + 1 dφ 2 dφ 2 u2 + Φ h 2 = 0 (1.53) and integrating over φ we have 1 2 ( ) 2 du + 1 dφ 2 u2 + Φ h 2 = constant = E h 2 (1.54) 49 / 59

62 Orbital periods Example and using h = r 2 φ E = r 4 φ 2 ( ) 2 du dφ 2 r 2 φ2 + Φ(r) = r 4 ( ) 2 du dt 2 r 2 φ 2 + Φ(r) = r 4 ( ) 2 du 2 dr ṙ r 2 φ2 + Φ(r) E h 2 = 1 2 Energy ( ) 2 du dφ u2 + Φ h 2 = 1 2ṙ r 2 φ2 + Φ(r) (1.55) i.e. we can show that the constant E we introduced is the energy per unit mass. 50 / 59

63 Orbital periods Example Peri and Apo ( ) 2 E h 2 = 1 du 2 dφ u2 + Φ h 2 For bound orbits, the limiting values of u (or r) occur where du dφ = 0, i.e. where u 2 = from (1.54). This has two roots, u 1 = 1 r 1 and u 2 = 1 r 2 For r 1 < r 2, where r 1 is the pericentre, r 2 the apocentre 2E 2Φ(u) h 2 (1.56) this is not obvious, since Φ is not defined, but it can be proved - it is an Example! 51 / 59

64 Orbital periods Example Orbital periods Radial motion The radial period T r is defined as the time to go from r 2 r 1 r 2. Now take (1.55) and re-write: E = 1 2 ṙ r 2 φ 2 + Φ(r) ( ) 2 dr = 2(E Φ(r)) h2 dt r 2 (1.57) where we used h = r 2 φ to eliminate φ So dr dt = ± 2(E Φ(r)) h2 r 2 (1.58) (two signs - ṙ can be either > 0 or < 0, and ṙ = 0 at r 1 & r 2. Then T r = r2 dt r2 odt = 2 r 1 dr dr = 2 dr (1.59) r 1 2(E Φ(r)) h2 r 2 52 / 59

65 Orbital periods Example Orbital periods Radial motion The radial period T r is defined as the time to go from r 2 r 1 r 2. Now take (1.55) and re-write: E = 1 2 ṙ r 2 φ 2 + Φ(r) ( ) 2 dr = 2(E Φ(r)) h2 dt r 2 (1.57) where we used h = r 2 φ to eliminate φ So dr dt = ± 2(E Φ(r)) h2 r 2 (1.58) (two signs - ṙ can be either > 0 or < 0, and ṙ = 0 at r 1 & r 2. Then T r = r2 dt r2 odt = 2 r 1 dr dr = 2 dr (1.59) r 1 2(E Φ(r)) h2 r 2 52 / 59

66 Orbital periods Azimuthal motion Orbital periods Example If travelling from r 2 r 1 r 2 φ is increased by an amount so φ = odφ = 2 r2 φ = 2h r 1 r2 r 1 dφ dr dr = 2 dr r2 r 1 dφ dt dt dr (1.60) dr r 2 2(E Φ(r)) h2 r 2 (1.61) 53 / 59

67 Precession of the orbit Orbital periods Example For a given orbit, the time taken to go around once (i.e. 0 2π) depends in general on where you start, so the azimuthal period is not well defined. Instead use the mean angular velocity ω = φ/t r to obtain a mean azimuthal period T φ, so T φ = 2π/ ω T φ = 2π φ T r is the mean time to go around once. Note that unless φ/2π is a rational number the orbit is not closed. 54 / 59

68 Orbital periods Example Precession of the orbit For Keplerian orbit φ = 2π T r = T φ. In one period T r the apocentre (or pericentre) advances by an angle φ 2π. i.e.the orbit shifts in azimuth at an average rate given by the mean precession rate Ω p = φ 2π T r rad s 1 (1.62) Thus the precession period is T p = 2π = T r Ω φ p 2π 1 (1.63) This precession is in the sense opposite to the rotation of the star In the special case of a Keplerian orbit φ = 2π T φ = T r and Ω p = 0, i.e. orbits are closed and do not precess. Otherwise general orbit is a rosette between r 1 & r 2. This allows us to visualize how we can build a galaxy out of stars on different orbits. 55 / 59

69 Precession of the orbit Orbital periods Example 56 / 59

70 Orbital periods Example T r for the Keplerian case Φ(r) = GM r We have equation (1.59) r2 dr T r = 2 r 1 2(E Φ(r)) h2 r 2 Now r 1 & r 2 are determined from ṙ = 0, i.e. Example 2(E Φ(r)) h2 r 2 = 0 (1.64) 2E + 2GM r r 1 r 2 = h2 2E ; (remember E < 0 for a bound orbit). h2 r 2 = 0 (1.65) r 2 + GM E r h2 2E = 0 (1.66) (r r 1 )(r r 2 ) = 0 (1.67) r 1 + r 2 = GM E (1.68) 57 / 59

71 Orbital periods Example T r for the Keplerian case Φ(r) = GM r We have equation (1.59) r2 dr T r = 2 r 1 2(E Φ(r)) h2 r 2 Now r 1 & r 2 are determined from ṙ = 0, i.e. Example 2(E Φ(r)) h2 r 2 = 0 (1.64) 2E + 2GM r r 1 r 2 = h2 2E ; (remember E < 0 for a bound orbit). h2 r 2 = 0 (1.65) r 2 + GM E r h2 2E = 0 (1.66) (r r 1 )(r r 2 ) = 0 (1.67) r 1 + r 2 = GM E (1.68) 57 / 59

72 Orbital periods Example Rewrite (1.59) as r2 T r = 2 r 1 rdr 2E(r r1 )(r r 2 ) = 2 r2 2 E r 1 Example rdr (r2 r)(r r 1 ) (1.69) if r 1 < r < r 2. This is another of those integrals. If R = a + bx + cx 2 = r 2 + (r 1 + r 2 )r r 1 r 2 and = 4ac b 2 which becomes, using the variables here, = (r 1 r 2 ) 2 then xdx R = b ( ) 1 2cx + b sin 1 R c 2c c for c < 0 and < 0 (See G&R and 2.264). 58 / 59

73 Orbital periods Example The first term is 0 at r 1 and r 2 (R = 0 there), so [ ] [ ( 2 r1 + r T r = 2 sin 1 2r2 + r 1 + r 2 2 E 2 r 1 r 2 = = [ 2 r1 + r 2 2 E 2 2 GM [ π ] 2 E 2( E) 2 ( π 2 ) = 2πGM ( 2E) 3 2 ] [sin 1 (1) sin 1 ( 1)] Example ) ( )] sin 1 2r1 + r 1 + r 2 r 1 r 2 (1.70) 59 / 59