1. Gravitational Dynamics

2. Gravitational Dynamics can be applied to: Two body systems:binary stars Planetary Systems Stellar Clusters:open & globular Galactic Structure:nuclei/bulge/disk/halo Clusters of Galaxies The universe:large scale structure

3. Syllabus Phase Space Fluid f(x,v) –Eq n of motion –Poisson’s equation Stellar Orbits –Integrals of motion (E,J) –Jeans Theorem Spherical Equilibrium –Virial Theorem –Jeans Equation Interacting Systems –Tides  Satellites  Streams –Relaxation  collisions

4. How to model motions of 10 10 stars in a galaxy? Direct N-body approach (as in simulations) –At time t particles have (m i,x i,y i,z i,vx i,vy i,vz i ), i=1,2,...,N (feasible for N<<10 6). Statistical or fluid approach (N very large) –At time t particles have a spatial density distribution n(x,y,z)*m, e.g., uniform, –at each point have a velocity distribution G(vx,vy,vz), e.g., a 3D Gaussian.

5. N-body Potential and Force In N-body system with mass m 1 …m N, the gravitational acceleration g(r) and potential φ(r) at position r is given by: r 12 RiRi r mimi

6. Eq. of Motion in N-body Newton’s law: a point mass m at position r moving with a velocity dr/dt with Potential energy Φ(r) =mφ(r) experiences a Force F=mg, accelerates with following Eq. of Motion :

7. Orbits defined by EoM & Gravity Solve for a complete prescription of history of a particle r(t) E.g., if G=0  F=0, Φ(r)=cst,  dx i /dt = vx i =c i  x i (t) =c i t +x 0, likewise for y i,z i (t) –E.g., relativistic neutrinos in universe go straight lines Repeat for all N particles.  N-body system fully described

8. Example: 5-body rectangle problem Four point masses m=3,4,5 at rest of three vertices of a P-triangle, integrate with time step=1 and ½ find the positions at time t=1.

9. Star clusters differ from air: Size doesn’t matter: –size of stars<<distance between them –  stars collide far less frequently than molecules in air. Inhomogeneous In a Gravitational Potential φ(r) Spectacularly rich in structure because φ(r) is non-linear function of r

10. Why Potential φ(r) ? More convenient to work with force, potential per unit mass. e.g. KE  ½v 2 Potential φ(r) is scaler, function of r only, –Easier to work with than force (vector, 3 components) –Simply relates to orbital energy E= φ(r) +½v 2

11. Example: energy per unit mass The orbital energy of a star is given by: 0 since and 0 for static potential. So orbital Energy is Conserved in a static potential.

12. Example: Force field of two-body system in Cartesian coordinates

13. Example: Energy is conserved The orbital energy of a star is given by: 0 since and 0 for static potential. So orbital Energy is Conserved in a static potential.

14. A fluid element: Potential & Gravity For large N or a continuous fluid, the gravity dg and potential dφ due to a small mass element dM is calculated by replacing m i with dM: r 12 R r dM d3Rd3R

15. Potential in a galaxy Replace a summation over all N-body particles with the integration: Remember dM=ρ(R)d 3 R for average density ρ(R) in small volume d 3 R So the equation for the gravitational force becomes: RRiRRi

16. Poisson’s Equation Relates potential with density Proof hints:

17. Gauss’s Theorem Gauss’s theorem is obtained by integrating poisson’s equation: i.e. the integral,over any closed surface, of the normal component of the gradient of the potential is equal to 4  G times the Mass enclosed within that surface.

18. Poisson’s Equation Poissons equation relates the potential to the density of matter generating the potential. It is given by:

19. Laplacian in various coordinates

20. Poisson’s eq. in Spherical systems Poisson’s eq. in a spherical potential with no θ or Φ dependence is:

21. Proof of Poissons Equation Consider a uniform distribution of mass of density ρ. r g

22. Take d/dr and multiply r 2  Take d/dr and divide r 2 

23. From Density to Mass M(r) M(r+dr)

24. From Potential to Density Use Poisson’s Equation The integrated form of Poisson’s equation is given by: From Gravitational Force to Potential

25. More on Spherical Systems Newton proved 2 results which enable us to calculate the potential of any spherical system very easily. NEWTONS 1 st THEOREM:A body that is inside a spherical shell of matter experiences no net gravitational force from that shell NEWTONS 2 nd THEOREM:The gravitational force on a body that lies outside a closed spherical shell of matter is the same as it would be if all the matter were concentrated at its centre.

26. Circular Velocity CIRCULAR VELOCITY= the speed of a test particle in a circular orbit at radius r. For a point mass:For a homogeneous sphere

27. Escape Velocity ESCAPE VELOCITY= velocity required in order for an object to escape from a gravitational potential well and arrive at  with zero KE. It is the velocity for which the kinetic energy balances potential. -ve

28. Example:Single Isothermal Sphere Model For a SINGLE ISOTHERMAL SPHERE (SIS) the line of sight velocity dispersion is constant. This also results in the circular velocity being constant (proof later). The potential and density are given by:

29. Proof: Density  r -2 n=-2 Log(  ) Log(r)

30. Proof: Potential We redefine the zero of potential  If the SIS extends to a radius r o then the mass and density distribution look like this: M r  r roro roro

31. Beyond r o : We choose the constant so that the potential is continuous at r=r o. r  logarithmic  r -1

32. So:

33. Plummer Model PLUMMER MODEL=the special case of the gravitational potential of a galaxy. This is a spherically symmetric potential of the form: Corresponding to a density: which can be proved using poisson’s equation.

34. The potential of the plummer model looks like this:  r 

35. Since, the potential is spherically symmetric g is also given by:  The density can then be obtained from: dM is found from the equation for M above and dV=4  r 2 dr. This gives (as before from Poisson’s)

36. Isochrone Potential We might expect that a spherical galaxy has roughly constant  near its centre and it falls to 0 at sufficiently large radii. i.e. A potential of this form is the ISOCHRONE POTENTIAL.

37. Orbits in Spherical Potentials The motion of a star in a centrally directed field of force is greatly simplified by the familiar law of conservation of angular momentum. Keplers 3 rd law pericentre apocentre

38.  eff

39. at the PERICENTRE and APOCENTRE There are two roots for One of them is the pericentre and the other is the apocentre. The RADIAL PERIOD T r is the time required for the star to travel from apocentre to pericentre and back. To determine T r we use:

40. The two possible signs arise because the star moves alternately in and out. In travelling from apocentre to pericentre and back, the azimuthal angle  increases by an amount:

41. The AZIMUTHAL PERIOD is In general will not be a rational number. Hence the orbit will not be closed. A typical orbit resembles a rosette and eventually passes through every point in the annulus between the circle of radius r p and r a. Orbits will only be closed if is an integer.

42. ?????????????Graphs The velocity of the star is slower at apocentre due to the conservation of angular momentum. ??????????????????????????

43. Fluid approach:Phase Space Density PHASE SPACE DENSITY:Number of stars per unit volume per unit velocity volume f(x,v) (all called Distribution Function DF). The total number of particles per unit volume is given by:

44. E.g., air particles with Gaussian velocity (rms velocity = σ in x,y,z directions): The distribution function is defined by: mdN=f(x,v)d 3 xd 3 v where dN is the number of particles per unit volume with a given range of velocities. The mass distribution function is given by f(x,v).

45. The total mass is then given by the integral of the mass distribution function over space and velocity volume: Note:in spherical coordinates d 3 x=4πr 2 dr The total momentum is given by:

46. The mean velocity is given by:

47. Example:molecules in a room: These are gamma functions

48. Gamma Functions:

49. Liouvilles Theorem We previously introduced the concept of phase space density. The concept of phase space density is useful because it has the nice property that it is incompessible for collisionless systems. A COLLISIONLESS SYSTEM is one where there are no collisions. All the constituent particles move under the influence of the mean potential generated by all the other particles. INCOMPRESSIBLE means that the phase-space density doesn’t change with time.

50. Consider Nstar identical particles moving in a small bundle through spacetime on neighbouring paths. If you measure the bundles volume in phase space (Vol=Δ x Δ p ) as a function of a parameter λ (e.g., time t) along the central path of the bundle. It can be shown that: df/dt=0! It can be seen that the region of phase space occupied by the particle deforms but maintains its area. The same is true for y-p y and z-p z. This is equivalent to saying that the phase space density f=Nstars/Vol is constant.  df/dt=0! pxpx x p x x

51. DF & Integrals of motion If some quantity I(x,v) is conserved i.e. We know that the phase space density is conserved i.e Therefore it is likely that f(x,v) depends on (x,v) through the function I(x,v), so f=f(I(x,v)).

52. Jeans theorem For most stellar systems the DF depends on (x,v) through generally three integrals of motion (conserved quantities), I i (x,v),i=1..3  f(x,v) = f(I 1 (x,v), I 2 (x,v), I 3 (x,v)) E.g., in Spherical Equilibrium, f is a function of energy E(x,v) and ang. mom. vector L(x,v).’s amplitude and z-component

53. EoM  Jeans eq.

54. Spherical Equilibrium System Described by potential φ(r) SPHERICAL: density ρ(r) depends on modulus of r. EQUILIBRIUM:Properties do not evolve with time.

55. Anisotropic DF f(E,L,Lz). Energy is conserved as: Angular Momentum Vector is conserved as: DF depends on Velocity Direction through L=r X v

56. Proof: Angular Momentum is Conserved  Sincethen the force is in the r direction.  both cross products on the RHS = 0. So Angular Momentum L is Conserved in Spherical Isotropic Self Gravitating Equilibrium Systems. Alternatively:  =r×F & F only has components in the r direction   =0 so

57. Proof: Energy is conserved The total energy of an orbit is given by: 0 since and 0 for equilibrium systems. So Energy is Conserved.

58. Linear Momentum The change in linear momentum with time is given by: Since P is not conserved. However, when you sum up over all stars, linear momentum is conserved.

59. Spherical Isotropic Self Gravitating Equilibrium Systems ISOTROPIC:The distribution function only depends on the modulus of the velocity rather than the direction. Note:the tangential direction has  and  components

60. Isotropic DF f(E) In a static potential the energy of an particle is conserved. So,if we write f as a function of E then it will agree with the statement: Note:E=energy per unit mass For incompressible fluids

61. So: E=cst since For a bound equilibrium system f(E) is positive everywhere (can be zero) and is monotonically decreasing.

62. SELF GRAVITATING:The masses are kept together by their mutual gravity. In non self gravitating systems the density that creates the potential is not equal to the density of stars. e.g a black hole with stars orbiting about it is not self gravitating.

63. Eddington Formulae EDDINGTON FORMULAE:These can be used to get the density as a function of r from the energy density distribution function f(E).

64. Proof of the 1 st Eddington Formula

65. So, from a given distribution function we can compute the spherical density.

66. Relation Between Pressure Gradient and Gravitational Force Pressure is given by:

68. So, this gives: This relates the pressure gradient to the gravitational force. This is the JEANS EQUATION. Note:  2 =P Note:-ve sign has gone since we reversed the limits.

69. g  (r)  (r)  2  (E) v esc M So, gravity, potential, density and Mass are all related and can be calculated from each other by several different methods.

70. Proof: Situation where V c 2 =const is a Singular Isothermal Sphere From Previously: Conservation of momentum gives:

71.  

72.  Since the circular velocity is independent of radius then so is the velocity dispersion  Isothermal.

73. Finding the normalising constant for the phase space density If we assume the phase space density is given by: 

74. We can then find the normalizing constant so that  (r) is reproduced. Note: you want to integrate f(E) over all energies that the star can have I.e. only energies above the potential  We are integrating over stars of different velocities ranging from 0 to .

75. One way is to stick the velocity into the distribution function: Using the substitution gives:

76. Now  can also be found from poissons equation. Substituting in  from before gives: Equating the r terms gives: as before.

77. Flattened Disks Here the potential is of the form  (R,z). No longer spherically symmetric. Now it is Axisymmetric

78. Orbits in Axisymmetric Potentials We employ a cylindrical coordinate system (R, ,z) centred on the galactic nucleus and align the z axis with the galaxies axis of symmetry. Stars confined to the equatorial plane have no way of perceiving that the potential is not spherically symmetric.  Their orbits will be identical to those in spherical potentials. R of a star on such an orbit oscillates around some mean value as the star revolves around the centre forming a rosette figure.

79. Reducing the Study of Orbits to a 2D Problem This is done by exploiting the conservation of the z component of angular momentum. Let the potential which we assume to be symmetric about the plane z=0, be  (R,z). The general equation of motion of the star is then: The acceleration in cylindrical coordinates is: Motion in the meridional plane

80. The component of angular momentum about the z- axis is conserved. If  has no dependence on  then the azimuthal angular momentum is conserved (r  F=0). Eliminating in the energy equation using conservation of angular momentum gives: Specific energy density in 3D  eff

81. Thus, the 3D motion of a star in an axisymmetric potential  (R,z) can be reduced to the motion of a star in a plane. This (non uniformly) rotating plane with cartesian coordinates (R,z) is often called the MERIDIONAL PLANE.  eff (R,z) is called the EFFECTIVE POTENTIAL.

82. Minimum in  eff The minimum in  eff has a simple physical significance. It occurs where: (2) is satisfied anywhere in the equatorial plane z=0. (1) is satisfied at radius R g where This is the condition for a circular orbit of angular speed

83. So the minimum in  eff occurs at the radius at which a circular orbit has angular momentum L z. The value of  eff at the minimum is the energy of this circular orbit. R  eff E  R cir

84. The angular momentum barrier for an orbit of energy E is given by The effective potential cannot be greater than the energy of the orbit. The equations of motion in the 2D meridional plane then become:.

85. The effective potential is the sum of the gravitational potential energy of the orbiting star and the kinetic energy associated with its motion in the  direction. Any difference between  eff and E is simply kinetic energy of the motion in the (R,z) plane. Since the kinetic energy is non negative, the orbit is restricted to the area of the meridional plane satisfying. The curve bounding this area is called the ZERO VELOCITY CURVE since the orbit can only reach this curve if its velocity is instantaneously zero.

86. When the star is close to z=0 the effective potential can be expanded to give Can generally be ignored since the gravitational force is only in the z direction close to the equatorial plane. 2 So, the orbit is oscillating in the z direction.

87. The orbits are bound between two radii (where the effective potential equals the total energy) and oscillates in the z direction. ????????graph of z vs R??????????

88. If the energy of the orbit is reduced the two points between which the orbit is bound eventually become one. You then get no radial oscillation. You have circular orbits in the plane of the galaxy. This is one of the closed orbits in an axisymmetric potential and has the property that. (Minimum in effective potential.) Centrifugal Force Gravitational force in radial direction

89. The following figure shows the total angular momentum along the orbit. ?????????????????????????? Angular momentum is not quite conserved but almost. This is because the star picks up angular momentum as it goes towards the plane and vice versa. These orbits can be thought of as being planar with more or less fixed eccentricity. The approximate orbital planes have a fixed inclination to the z axis but they precess about this axis.

90. Orbits in Planar Non-Axisymmetrical Potentials Here the angular momentum is not exactly conserved. There are two main types of orbit –BOX ORBITS –LOOP ORBITS

91. LOOP ORBITS Star rotates in a fixed sense about the centre of the potential while oscillating in radius Star orbits between allowed radii given by its energy. There are two periods associated with the orbit: –Period of the radial oscillation –Period of the star going around 2  The energy is generally conserved and determines the outer radius of the orbit. The inner radius is determined by the angular momentum. pericentre apocentre

92. Box Orbits Have no particular sense of circulation about the centre. They are the sum of independent harmonic motions parallel to the x and y axes. In a logarithmic potential of the form box orbits will occur when R >R c.

93. JEANS EQUATION The jeans equation for oblate rotator: –a steady-state axisymmetrical system in which  2 is isotropic and the only streaming motion is azimuthal rotation:

94. The velocity dispersions in this case are given by: If we know the forms of  (R,z) and  (R,z) then at any radius R we may integrate the Jeans equation in the z direction to obtain  2.

95. Obtaining  2 Inserting this into the jeans equation in the R direction gives:

96. Example: suppose

98. Tidal Stripping TIDAL RADIUS:Radius within which a particle is bound to the satellite rather than the host system. Consider a satellite of mass profile m s (R) moving in a spherical potential  (r) made from a mass profile M(r). r R V(r)

99. The condition for a particle to be bound to the satellite rather than the host system is: Differential (tidal) force on the particle due to the host galaxy Force on particle due to satellite

100. So, Therefore the formula for Tidal Radius is: The tidal radius is smallest at pericentre where r is smallest. The inequality can be written in terms of the mean densities. The satellite mass is not constant as the tidal radius changes.

101. Lagrange Points There is a point between two bodies where a particle can belong to either one of them. This point is known as the LAGRANGE POINT. A small body orbiting at this point would remain in the orbital plane of the two massive bodies. The Lagrange points mark positions where the gravitational pull of the two large masses precisely cancels the centripetal acceleration required to rotate with them. At the lagrange points:

102. Effective Force of Gravity A particle will experience gravity due to the galaxy and the satellite along with a centrifugal force and a coriolis term. The effective force of gravity is given by: The acceleration of the particle is given by: Coriolis term Centrifugal term

103. Jakobi’s Energy The JAKOBI’S ENERGY is given by: Jakobi’s energy is conserved because

104. For an orbit in the plane (r perpendicular to  )

105. E J is also known as the Jakobi Integral. Since v 2 is always positive a star whose Jakobi integral takes the value E J will never tresspass into a region where  eff (x)>E J. Consequently the surface  eff (x)>E J, which we call the zero velocity surface for stars of Jakobi Integral E J, forms an impenetrable wall for such stars.

106. By taking a Taylor Expansion r=r s, you end up with: Where we call the radius r J the JAKOBI LIMIT of the mass m. This provides a crude estimate of the tidal radius r t.

107. Dynamical Friction DYNAMICAL FRICTION slows a satellite on its orbit causing it to spiral towards the centre of the parent galaxy. As the satellite moves through a sea of stars I.e. the individual stars in the parent galaxy the satellites gravity alters the trajectory of the stars, building up a slight density enhancement of stars behind the satellite The gravity from the wake pulls backwards on the satellites motion, slowing it down a little

108. The satellite loses angular momentum and slowly spirals inwards. This effect is referred to as “dynamical friction” because it acts like a frictional or viscous force, but it’s pure gravity.

109. More massive satellites feel a greater friction since they can alter trajectories more and build up a more massive wake behind them. Dynamical friction is stronger in higher density regions since there are more stars to contribute to the wake so the wake is more massive. For low v the dynamical friction increases as v increases since the build up of a wake depends on the speed of the satellite being large enough so that it can scatter stars preferentially behind it (if it’s not moving, it scatters as many stars in front as it does behind). However, at high speeds the frictional force  v -2, since the ability to scatter drops as the velocity increases. Note: both stars and dark matter contribute to dynamical friction

110. The dynamical friction acting on a satellite of mass m s moving at v s kms -1 in a sea of particles with gaussian velocity distribution is

111. For an isotropic distribution of stellar velocities this is: i.e. only stars moving slower than M contribute to the force. This is usually called the Chandrasekhar Dynamical Friction Formula. For sufficiently small v M we may replace f(v M ) by f(0) to give: For a sufficiently large v M, the integral converges to a definite limit and the frictional force therefore falls like v M -2.

112. When there is dynamical friction there is a drag force which dissipates angular momentum. The decay is faster at pericentre resulting in the staircase like appearance. ??????????????????????????????????

113. As the satellite moves inward the tidal force becomes greater so the tidal radius decreases and the mass will decay. ???????????????????????????????????????

114. Scalar Virial Theorem The Scalar Virial Theorem states that the kinetic energy of a system with mass M is just where is the mean-squared speed of the system’s stars. Hence the virial theorem states that Virial

115. Equation of motion: This is Tensor Virial Theorem

116. E.g. So the time averaged value of v 2 is equal to the time averaged value of the circular velocity squared.

117. In a spherical potential So =0 since the average value of xy will be zero.  =0

118. Adiabatic Invariance Suppose we have a sequence of potentials  p (|r|) that depends continuously on the parameter P(t). P(t) varies slowly with time. For each fixed P we would assume that the orbits supported by  p (r) are regular and thus phase space is filled by arrays of nested tori on which phase points of individual stars move. Suppose

119. The orbit energy of a test particle will change. Suppose The angular momentum J is still conserved because r  F=0. E.g. circular orbit in a flat potential: Initial Final Let P(t)=V cir 

120. So, if V cir increases by a factor of 2 then the radius of the orbit would decrease by a factor of 2 and the period would decrease by a factor of 4. E.g. in a potential In general the orbital eccentricity is conserved e.g. a circular orbit will stay circular.

121. In general, two stellar phase points that started out on the same torus will move onto two different tori. However, if P is changed very slowly compared to all characteristic times associated with the motion on each torus, all phase points that are initially on a given torus will be equally affected by the variation in P Any two stars that are on a common orbit will still be on a common orbit after the variation in P is complete. This is ADIABATIC INVARIANCE.