AST 1420 Galactic Structure and Dynamics Today: disks! NGC 5907 - - PowerPoint PPT Presentation

▶
ast 1420 galactic structure and dynamics today disks
SMART_READER_LITE
LIVE PREVIEW

AST 1420 Galactic Structure and Dynamics Today: disks! NGC 5907 - - PowerPoint PPT Presentation

Aug 26, 2023 161 likes •829 views

AST 1420 Galactic Structure and Dynamics Today: disks! NGC 5907 M31 Today: disks! Outline Simple gravitational potentials for disks Realistic potentials for disks Orbits in axisymmetric potentials Close-to-circular orbits

slide-1
SLIDE 1

AST 1420 Galactic Structure and Dynamics

slide-2
SLIDE 2

Today: disks!

NGC 5907 M31

slide-3
SLIDE 3

Today: disks!

slide-4
SLIDE 4

Outline

  • Simple gravitational potentials for disks
  • Realistic potentials for disks
  • Orbits in axisymmetric potentials
  • Close-to-circular orbits (e.g., stars near the Sun)
slide-5
SLIDE 5

Computing the gravitational potential for a disk

  • Simple: just solve the Poisson equation!
  • Newton’s theorems don’t apply
slide-6
SLIDE 6

Razor-thin disks

  • ‘Extreme’ assumption: disk is infinitely thin
  • Integrate Poisson equation
  • Divergence theorem
  • For small volume
slide-7
SLIDE 7
  • At any z > 0: point mass at (0,-a)
  • At any z < 0: point mass at (0,a)
  • For any z =/= 0 —> density = 0 —> razor-thin
  • At z=0 ~ Plummer —> non-zero density

Example: Kuzmin disk

slide-8
SLIDE 8

Example: Kuzmin disk

slide-9
SLIDE 9

Thick disks: Miyamoto- Nagai

  • Razor-thin is not realistic for orbits in disks
  • Can thicken Kuzmin by replacing |z| —> √[z2 + b2]
  • Miyamoto-Nagai:
  • b —> 0: Kuzmin
  • a —> 0: Plummer (spherical)
slide-10
SLIDE 10

Thick disks: Miyamoto- Nagai

slide-11
SLIDE 11

Thick disks: Miyamoto- Nagai

Potential is much less flattened:

slide-12
SLIDE 12
  • Advantages:
  • Analytical formula —> fast for, e.g., orbit integration
  • Can vary b/a to get different ‘thicknesses’
  • Disadvantages:
  • Large R ~ R-3 —> not like a realistic disk, too much

density at large R

  • Similarly, vertical profile not realistic

Thick disks: Miyamoto- Nagai

slide-13
SLIDE 13

Miyamoto-Nagai rotation curve

slide-14
SLIDE 14

Flattened + flat rotation curve

  • Spherical model:
  • Has vc(r) = constant
  • General flattening strategy: r —> m = √[R2 + z2/q2]
  • Has vc(R) = constant at z=0
slide-15
SLIDE 15

Flattened logarithmic potential

  • Potential for q=0.7 is slightly flattened:
slide-16
SLIDE 16

Flattened logarithmic potential

  • But density becomes negative around R=0!
slide-17
SLIDE 17

Flattened logarithmic potential: density

  • When q < 1/√2: density is negative at R=0
  • and everywhere between
  • Thus, you cannot flatten a logarithmic potential too much

without creating negative densities

  • General problem when flattening a potential using r —> m

= √[R2 + z2/q2] [e.g., often used to get flattened DM halos]

slide-18
SLIDE 18

Potentials for disk densities

slide-19
SLIDE 19

Poisson equation for disks

  • Write density as
  • Poisson equation best written in cylindrical

coordinates

  • Take Fourier transform of phi part
slide-20
SLIDE 20

Poisson equation for disks

  • or …
  • Similar to the Bessel differential equation:
slide-21
SLIDE 21

Poisson equation for disks

  • Solutions of the form
slide-22
SLIDE 22

Poisson equation for axisymmetric razor-thin disks

  • Solution:
  • With 𝛵
slide-23
SLIDE 23

Poisson equation for axisymmetric razor-thin disks

  • If we can decompose
  • Then the potential is
  • Decomposition from the Fourier-Bessel theorem
slide-24
SLIDE 24

Circular velocity for axisymmetric razor-thin disks

slide-25
SLIDE 25

Example: Mestel disk

  • Razor-thin disk with surface density
  • Hankel transform:
  • Potential:
slide-26
SLIDE 26

Example: Mestel disk

  • Enclosed mass:
  • Same as for spherical potential! Coincidence!
slide-27
SLIDE 27

Example: Exponential disk

  • Observed disks have exponential(-ish) surface

density profiles:

  • Hankel transform:
  • Potential:
slide-28
SLIDE 28

Example: Exponential disk

slide-29
SLIDE 29

Example: Exponential disk

slide-30
SLIDE 30

Potentials with finite thickness

  • In general difficult, but galaxies overall well

approximated as

  • Then each layer gives rise to the ~same potential,

but shifted in z

  • Full potential adds up contribution from each layer
slide-31
SLIDE 31

Example: 
 Double-exponential disk

  • Density for which both radial and vertical profile are

exponential

  • Full potential + forces: one-dimensional numerical

integrals

slide-32
SLIDE 32

Double-exponential disk: rotation curve

  • Rotation curve becomes:
slide-33
SLIDE 33

Double-exponential disk: rotation curve

slide-34
SLIDE 34

Miyamoto-Nagai rotation curve

slide-35
SLIDE 35

Orbits in axisymmetric disks

slide-36
SLIDE 36

Orbits in axisymmetric disks

  • Approximate model for disk galaxy:
  • Flattened axisymmetric disk
  • Symmetric around z=0
  • Often use Miyamoto-Nagai for computational convenience
  • E.g., galpy’s Milky-Way model
  • Miyamoto-Nagai disk with scale length 3 kpc, scale height

280 pc

  • NFW halo
  • Spherical bulge with exponential cut-off
slide-37
SLIDE 37
  • Lagrangian in cylindrical coordinates
  • With conjugate momenta
  • Hamiltonian

Orbits in cylindrical geometry

slide-38
SLIDE 38

Orbits in cylindrical geometry

  • z component of the angular momentum is

conserved

  • Hamiltonian becomes
  • with
  • Effectively a two (four) dimensional system in (R,z)

—> meridional plane

slide-39
SLIDE 39

Motion in the meridional plane

  • Equations of motion
  • Coupled oscillations in R and z
  • No analytical solutions
slide-40
SLIDE 40

Motion in the meridional plane: effective potential

slide-41
SLIDE 41

http://astro.utoronto.ca/~bovy/AST1420/orbits/lec5-orbitexample1.html

slide-42
SLIDE 42

Motion in the meridional plane

slide-43
SLIDE 43

http://astro.utoronto.ca/~bovy/AST1420/orbits/lec5-orbitexample2.html

slide-44
SLIDE 44

Motion in the meridional plane

slide-45
SLIDE 45

Motion in the meridional plane

slide-46
SLIDE 46

Isolating integrals of the motion

  • Time-independent potential: E conserved —> motion

restrained to ϕ < E

  • Spherical potential: L conserved —> motion restrained to (a)
  • rbital plane (b) vT = |L|/R (ϕeff < E)
  • Motion fills rest of the phase-space
slide-47
SLIDE 47

http://astro.utoronto.ca/~bovy/AST1420/orbits/lec5-orbitexample3.html

slide-48
SLIDE 48

“Third integral”

  • Fact that orbits in axisymmetric do not fully explore the area in

the meridional plane alllowed by ϕeff < E means there has to be an additional integral

  • No known classical integral (like E or Lz) —> non-classical ‘third’

integral

  • Will come back to this in a few lectures
  • Focus here on approximate understanding
slide-49
SLIDE 49

Separability of disk orbits

  • Orbits in disks are to a good approximation

independent oscillations in R and z (coupling is small)

  • Will be highly useful to understand equilibrium

models of disks

  • Write the potential as
slide-50
SLIDE 50

Motion in the meridional plane

slide-51
SLIDE 51

Separability of disk orbits

slide-52
SLIDE 52

Separability of disk orbits

slide-53
SLIDE 53

Separability of disk orbits

slide-54
SLIDE 54

Separability of disk orbits

slide-55
SLIDE 55

Motion in the meridional plane

slide-56
SLIDE 56

Separability of disk orbits

slide-57
SLIDE 57

more typical orbit…

slide-58
SLIDE 58

Separability of disk orbits

slide-59
SLIDE 59

Separability of disk orbits

slide-60
SLIDE 60

Separability of disk orbits

slide-61
SLIDE 61

Separability of disk orbits

  • Orbits in disks are to a good approximation

independent oscillations in R and z (coupling is small)

  • Will be highly useful to understand equilibrium models
  • f disks
  • Write the potential as
  • Hamiltonian then splits into two pieces, with separately

conserved energies

slide-62
SLIDE 62

Epicycle approximation for close-to-circular orbits

slide-63
SLIDE 63

Potential for close to circular

  • rbits
  • Circular orbit is the minimum of the effective

potential, can Taylor expand around the minimum

  • Motion is then explicitly two decoupled oscillators
slide-64
SLIDE 64

Frequencies

  • Frequencies of the oscillations:
  • +azimuthal Ω = vc(R) / R
  • Range:
slide-65
SLIDE 65

Vertical motion

  • Solution is sinusoidal
  • But bad approximation because disks are thin
  • Vertical decoupling is useful, vertical epicycles not

so much

slide-66
SLIDE 66

Radial motion

  • Radial oscillation around guiding-center radius:

radius of circular orbit with angular momentum

  • Azimuthal motion from conservation of angular

momentum

  • Subtracting out motion of guiding center, motion is

ellipse: epicycle

  • Axis ratio