AST 1420 Galactic Structure and Dynamics Presentations Week 11: - - PowerPoint PPT Presentation

AST 1420 Galactic Structure and Dynamics

Presentations

• Week 11: Nov. 24
• Each student presents on a topic for ~10 min.
• Encouraged to find your own topic in Galactic structure and

dynamics!

• Could be a survey and some results on a topic addressed by the

survey: e.g., Gaia and co-moving stars, ATLAS 3D integral-field- spectroscopy and the IMF, APOGEE and chemical evolution,

• Or a topic: e.g., rotation curves of low-surface brightness

galaxies, rotation curves at redshift ~ 2, the dynamics of the inner Milky Way, Schwarzschild modeling of galactic nuclei to constrain black holes, …

• Please email me with your proposed topic by Oct. 20

Assignment 2 due today!

So far

• Properties of spherical and mass distributions
• General properties of orbits; some orbits in disks
• Equilibrium of spherical and axisymmetric galactic

systems

• What about more complicated geometries?

Today

• Elliptical galaxies / dark-matter halos
• Potential of a spheroidal and ellipsoidal (triaxial) mass

distributions

• Surfaces of section as a way to study orbits
• Orbits in planar non-axisymmetric potentials (—> triaxial)
• Chaos and integrals of the motion
• Orbits in triaxial potentials

Elliptical galaxies

M87

NGC 4660

What is the intrinsic shape of elliptical galaxies?

• We only observe the projected shape
• For an individual elliptical galaxy, cannot tell

whether it is intrinsically axisymmetric or triaxial Credit: Chris Mihos

Dispersion vs rotation supported systems

• We have discussed multiple types of systems so far:
• Spherical systems: no net rotation, Jeans equation

relates mass to velocity dispersion —> dispersion supported

• Disk systems: high rotation velocity, can write down DF

for any razor-thin disk that only consists of circular orbits —> rotation supported

• In general, stellar systems are supported against

gravitational collapse by having (a) large velocity dispersion, (b) large rotation velocity, (c) some combination

• Starting point: assume elliptical galaxies are

axisymmetric, flattened through rotation

• E.g., look at Jeans equation from last week

What is the intrinsic shape of elliptical galaxies?

• Could be that dispersion tensor is isotropic, if mean

rotation is high enough

• Rotation would have to be few 100 km/s for large ellipticals

Rotation of elliptical galaxies

• Large ellipticals (M >~ 1012 Msun) do not rotate

much

• Therefore, must be dispersion supported
• Velocity dispersion must be anisotropic to support a non-

spherical, axisymmetric system

• If elliptical galaxies are non-spherical, maybe they are even

triaxial? (Binney 1978)

Shape of elliptical galaxies

• Thus, elliptical galaxies are at least axisymmetric —

> spheroids?

• Sky projection should give similar ellipses if shape

is constant with radius

Shape of elliptical galaxies

• Observed isophotes of large ellipticals twist
• Could be because the major-axis of a spheroidal

shell twists —> resulting model is non-axisymmetric

Credit: Kormendy

Shape of elliptical galaxies

• Alternative explanation: galaxy is a triaxial ellipsoid
• Density constant on shells

Credit: Kormendy

Shape of elliptical galaxies

• Isophotal twist can result from change in a/b, a/c,

but no change in orientation —> triaxial

Elliptical galaxies as triaxial mass distributions

• Isophote twists —> ellipticals cannot be exactly axisymmetric
• Distribution of observed axis ratios inconsistent with random

projection of intrinsically oblate or prolate distribution (e.g., Ryden 1996)

• Triaxial velocity ellipsoid can support triaxiality under self-gravity —

> kinematics then misaligned with photometric isophotes —>

• bserved for large ellipticals (e.g., Weijmans et al. 2014)
• From later: triaxial potential —> stable orbit looping around major

axis —> observationally shows up as minor-axis rotation

• Small amounts of minor-axis rotation are observed for ellipticals

(e.g., Franx et al. 1991 )

Low-mass ellipticals

• Lower-mass ellipticals (M <~ 1011.5 Msun) appear to

be almost axisymmetric:

• Kinematics aligned with photometry
• Relatively fast rotation (but still much support

from dispersion)

• Lower-mass ellipticals better represented by

axisymmetric model, but much puffier than the disks from last weeks

Dark-matter halos

Universal profile of dark matter halos

Navarro, Frenk, & White (1997)

• Numerical simulations of

formation of dark matter halos find universal profile: NFW

• Profile is the same shape

for all masses, but inner density varies —> lower mass halos are less dens

• All halos have inner

density cusp

ρ(r) = ρ0 r0 r (1 − r/r0)2

Shape of dark-matter halos

• Cosmological simulations of the formation
• f dark-matter halos (without baryonic

effects) find that halos are strongly triaxial (e.g., Frenk et al. 1988, Dubinski & Carlberg 1991)

• Structures can be stable over the age of

the Universe

• Growth of baryonic component reduces

triaxiality near the center and DM halo typically oblate

• But outer structure of halos likely quite

triaxial; direct measurements rare

• Important to understand, because shape

can be sensitive to DM microphysics

Dubinski & Carlberg (1991)

Potentials for triaxial mass distributions

Potentials for mildly- flattened axisymmetric mass distributions

Spheroidal and ellipsoidal shapes

a=b > c a=b < c a=/=b=/=c Spheroidal Ellipsoidal

• For

Potential of spheroidal system: oblate shell

• Let’s start by considering an oblate spheroidal shell
• q < 1; eccentricity
• As usual, we need to solve the Poisson equation

Solving partial differential equations: separation of variables

• Partial differential equations (PDEs) are difficult to

solve!

• Often possible to solve them using separation of

variables

• Write solution as product of functions of 1 variable,

PDE splits into set of independent ODEs

rΦ(x, y, z) = 0

Solving partial differential equations: separation of variables

• Example: Laplace equation in cartesian

coordinates

• Solution: ɸ(x,y,z) = X(x) Y(y) Z(z)

∂2[X(x) Y (y) Z(z)] ∂x2 + ∂2[X(x) Y (y) Z(z)] ∂y2 + ∂2[X(x) Y (y) Z(z)] ∂z2 = 0

1 X(x) ∂2X(x) ∂x2 + 1 Y (y) ∂2Y (y) ∂y2 + 1 Z(z) ∂2Z(z) ∂z2 = 0

• Or

Solving partial differential equations: separation of variables

1 X(x) ∂2X(x) ∂x2 + 1 Y (y) ∂2Y (y) ∂y2 + 1 Z(z) ∂2Z(z) ∂z2 = 0

1 X(x) ∂2X(x) ∂x2 + 1 Y (y) ∂2Y (y) ∂y2 = − 1 Z(z) ∂2Z(z) ∂z2

• Or
• Each side must be constant

1 X(x) ∂2X(x) ∂x2 + 1 Y (y) ∂2Y (y) ∂y2 = m 1 Z(z) ∂2Z(z) ∂z2 = −m

Solving partial differential equations: separation of variables

• Each side must be constant
• Again

1 X(x) ∂2X(x) ∂x2 = l 1 Y (y) ∂2Y (y) ∂y2 = m − l 1 X(x) ∂2X(x) ∂x2 + 1 Y (y) ∂2Y (y) ∂y2 = m 1 X(x) ∂2X(x) ∂x2 = m − 1 Y (y) ∂2Y (y) ∂y2

Laplace equation: separation of variables

• Useful when:
• All involved functions separate: for Poisson, when

the density separates

• We did this for disk mass distribution functions —>

Hankel transformation

• Useful for oblate shell if we can find coordinates in which

the Laplace equation separates

• Laplace equation separates in 13 different coordinate

systems (cartesian, cylindrical, spherical, …)

Oblate spheroidal coordinates

• Laplace equation separates in oblate spheroidal coordinates

(u,v,φ)

• Like cylindrical, but (R,z) replaced by (u,v)
• In (R,z): curves of constant u: half-ellipses with focus at

(R,z) = (Δ,0), eccentricity tanh(u)

• Curves of constant v: hyperbolae with same focus

Oblate spheroidal coordinates

• Oblate spheroidal shell at

m=m0 and 
 eccentricity 
 corresponds to constant u shell if

• Shell coordinate is then

Oblate spheroidal coordinates: Laplace equation

• Laplace equation can be separated in oblate spheroidal coordinates
• Potential of a shell that is only a function of u is therefore only a function of u

itself

• At large u, cosh u —> sinh u and Δ cosh u —> r 


—> oblate spheroidal coordinates become ~spherical coordinates

• Thus, at large distances from the shell, the potential becomes spherical
• The Laplacian is

Oblate spheroidal coordinates: Laplace equation

• We only care about u part for the separation of

variables approach

• Solution

Potential of an oblate shell

• Potential inside the shell is
• Potential outside the shell is
• Constants set by boundary conditions:
• Boundary at 0: force must be zero —> Ain = 0 —> inside ɸ constant
• Boundary at infinity: set to zero —> Bout = 0
• Boundary at the shell u=u0: Bin = Aout sin
• 1(1/cosh[u0])

Potential of an oblate shell

• A set by the mass of the shell
• Integrate Poisson equation over the entire shell in u and

φ and infinitesimal range in u around u0

Potential of an oblate shell

Potential of a prolate shell

• We first write down the Laplace equation in prolate spheroidal

coordinates and then solve it …

• Just kidding! Won’t go through this in detail
• But let’s take a look at prolate spheroidal coordinates

• Like for spherical mass: decompose into shells
• Compute potential of shell using previous framework
• Add up potentials from all shells
• Not as straightforward as it seems, because you need

different spheroidal coordinate systems for each shell (not a constant focus)

Potential of a mass distribution that is stratified on similar spheroids

Potential of a mass distribution that is stratified on similar spheroids

Potentials triaxial, ellipsoidal mass distributions

Potential of an ellipsoidal shell

• Shell at constant m
• Laplace equation actually separates in ellipsoidal coordinates

as well

• Bet you want to know what those are!

de Zeeuw (1985)

Potential of an ellipsoidal shell

Surfaces of section

Surfaces of section

• So far we have visualized orbits simply by plotting them in (x,y), (R,z),

(R,vR), …

• To better study the structure of orbits, a better visualization is useful
• Principle of surface of section: reduce dimensionality of orbit by one by
• nly showing the orbit when it intersects with a (hyper)-surface
• Useful for axisymmetric potentials, because they are
• effectively 4D due to conservation of Lz —> (R,z,vR,vz)
• 3D because of conservation of E —> (R,z,vR) and choose pos. vz
• and then 2D because of intersection with plane (e.g., z=0) —> (R,vR)

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

Motion in the meridional plane

Surface of section

• Would have expected this to fill 2D plane, maybe longer

integration is necessary?

Surface of section

• Still 1D —> there must be a 3rd integral for this orbit

• Useful to visualize the orbital structure of a galaxy
• For example: sequence of orbits with constant

(E,Lz) but different initial vR

Surfaces of section

Motion in the meridional plane

Motion in the meridional plane

Surface of section at constant (E,Lz)

Surface of section at constant energy

• Can more generally study what the orbits look like

at constant energy: choose intermediate vR

• Purple orbit is even more restrained, it doesn’t even fill an entire

closed curve! —> radial and vertical frequency must be commensurate, such that the orbit doesn’t fill (R,z) symmetrically

• Let’s look at this orbit in (R,z) and (vR,vz)

Surface of section: resonant

• rbits

Surface of section: resonant

• rbits

Orbits in 2D, non- axisymmetric potentials

Orbits in planar, non- axisymmetric potentials

• To start to understand how the orbital structure of

triaxial galaxies is different from that of axisymmetric galaxies, let’s look at 2D non-axisymmetric potentials

• Useful to work in 2D, because can still use a surface of

section:

• Energy conservation —> (x,y,vx)
• Surface of section (e.g., y=0) —> (x,vx)
• Can still easily visualize this

Simple 2D, non- axisymmetric potential

• Flattened logarithmic potential, with flattening in y

and core radius

• Let’s look at orbit with x < Rc
• http://astro.utoronto.ca/~bovy/AST1420/notes/notebooks/08.-

Triaxial-Mass-Distributions.html#galpy- smwzvjryywgkrnimblalukpu

• Orbit fills box and goes arbitrarily close to the center!

Box orbits

• Box orbits result because the potential is close to a

harmonic oscillator near the center

• For example, orbit with x << Rc

Box orbits

Loop orbits

• Let’s look at orbit with x > Rc
• http://astro.utoronto.ca/~bovy/AST1420/notes/notebooks/08.-

Triaxial-Mass-Distributions.html#galpy-jovkdpzxoioeohlfrlxmcfyq

• Orbit loops around the center like orbits in an axisymmetric

potential

Orbital structure

• As we go out in radius, orbits change from boxes to

loops

Orbital structure: surface of section

• Clearer what is going on using a surface of section
• Same sequence of orbits

Orbital structure: surface of section at constant energy

• Points are closed orbits: closed loop orbit —> parent of loop orbits
• Outermost always has y=0, vy=0: closed long-axis orbit —> parent of box orbits
• Both are stable closed orbits: surrounded by their children

More non-axisymmetric —> fewer loop orbits

• At a certain point, closed loop orbits ‘meet’ at x=0 and

loop orbits disappear —> replaced by closed short- axis box orbit

Chaos and integrals of the motion

Chaos

• So far, all orbits that we have encountered have been well-behaved
• All orbits in the planar, non-axisymmetric potential formed closed

curves —> second integral beside the energy (analogous to the third integral for axisymmetric systems)

• Orbits with 2 (in 2D) or 3 (in 3D) integrals of the motion are well-

behaved and oscillate quasi-periodically

• Orbits with fewer integrals of the motion can explore more

dimensions of phase space and ‘fill’ more space —> erratic, chaotic behavior

• Do all orbits in an axisymmetric potential have a third integral —>

no chaos?

Do all orbits in an axisymmetric potential have a third integral?

• Henon & Heiles (1964) set out to investigate this
• 3D axisymmetric potential —> effective 2D system

in the (R,z) meridional plane with effective potential ɸ(R,z) + Lz/2/R2

• Henon & Heiles: therefore study 2D, non-

axisymmetric potential! [exactly like 3D if Lz = 0]

• To answer the principled question, consider a

somewhat funny potential (but very famous now)

Henon & Heiles (1964) potential

Henon & Heiles (1964)

Henon & Heiles (1964) potential

• Surface of section in (y,vy) at E=1/12

Henon & Heiles (1964) potential

• Surface of section in (y,vy) at E=1/12
• Four closed orbits: 2 loops, 2 boxes

Henon & Heiles (1964) potential

• Surface of section in (y,vy) at E=1/12
• Four closed orbits: 2 loops, 2 boxes —> stable parents of orbits families

Henon & Heiles (1964) potential

• Surface of section in (y,vy) at E=1/12
• Purple curve: orbits at the edge of the four orbit families

Henon & Heiles (1964) potential

• Surface of section in (y,vy) at E=1/8

Henon & Heiles (1964) potential

• Surface of section in (y,vy) at E=1/8
• Purple orbit: non-regular: fills a 2D space! —> chaotic orbit

Henon & Heiles (1964) potential

• Surface of section in (y,vy) at E=1/8
• Similar stable closed orbits as before
• Islands around the areas around the

main closed orbits

• Centers of these islands are closed
• rbits themselves
• Center of the small, orange islands:

Henon & Heiles (1964) potential

• Surface of section in (y,vy) at E=1/8
• Orange islands are curious: they don’t close up
• Let’s zoom in on these parts of the surface of section
• Islands all the way down… —> chaos

Henon & Heiles (1964) potential

• Surface of section in (y,vy) at E=1/8
• Pink orbit: closed orbit in the middle of the chaotic sea
• Orbit integration of the same length for pink and purple

Do all orbits in an axisymmetric potential have a third integral?

No!

• Potentials for which all orbits have a third integral are called

integrable (e.g., the Kuzmin potential)

• Other potentials are non-integrable
• Non-integrable potentials may have large parts of phase

space that are integrable (where orbits have a third integral)

Do all orbits in galactic potentials have a third integral?

• For general galaxies difficult to determine
• Milky Way: solar neighborhood σz =/= σR
• If (E,Lz) were the only

integrals —> σz =/= σR
 because 
 E = 0.5[vR2+vz2]

• Thus, most orbits in the

Milky Way disk must have a third integral

Bovy et al. (2009)

Orbits in triaxial potentials

Orbits in triaxial potentials

• Qualitatively, orbits in

triaxial potentials are similar to those in planar, non- axisymmetric potentials:

• Family of box orbits
• Short-axis tube orbit (~

loop orbits)

• Inner and outer long-axis

tube orbit

• No stable orbit around

the intermediate axis

Statler (1987)

Orbital support for triaxiality

• Box orbits are very important for supporting the triaxiality of the

potential, especially near the center

• Chaos acts to make make the DF more uniform, because the

phase-space density should be ~constant along chaotic trajectories
 —> the velocity dispersion becomes isotropic and thus the support for triaxiality gets destroyed

• Chaos can be induced by central cusp, a central black hole, which

scatter the box orbits when they pass close to the center

• Similarly, growth of axisymmetric disk makes inner galaxy

~axisymmetric —> destroys support for DM box orbits —> DM halo becomes axisymmetric as well