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

What is the intrinsic shape of elliptical galaxies? • Starting point: assume elliptical galaxies are axisymmetric, flattened through rotation • E.g., look at Jeans equation from last week • 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 >~ 10 12 M sun ) 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 Credit: Kormendy • Could be because the major-axis of a spheroidal shell twists —> resulting model is non-axisymmetric

Shape of elliptical galaxies • Alternative explanation: galaxy is a triaxial ellipsoid • Density constant on shells

Shape of elliptical galaxies • Isophotal twist can result from change in a/b, a/c, but no change in orientation —> triaxial Credit: Kormendy

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 —> observed 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 <~ 10 11.5 M sun ) 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 • Numerical simulations of formation of dark matter halos find universal profile: NFW r 0 ρ ( r ) = ρ 0 r (1 − r/r 0 ) 2 • Profile is the same shape for all masses, but inner density varies —> lower mass halos are less dens • All halos have inner density cusp Navarro, Frenk, & White (1997)

Shape of dark-matter halos • Cosmological simulations of the formation of 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 Dubinski & Carlberg (1991) • Important to understand, because shape can be sensitive to DM microphysics

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

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 • For

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

Solving partial differential equations: separation of variables • Example: Laplace equation in cartesian coordinates r Φ ( x, y, z ) = 0 • Solution: ɸ (x,y,z) = X(x) Y(y) Z(z) ∂ 2 [ X ( x ) Y ( y ) Z ( z )] + ∂ 2 [ X ( x ) Y ( y ) Z ( z )] + ∂ 2 [ X ( x ) Y ( y ) Z ( z )] = 0 ∂ x 2 ∂ y 2 ∂ z 2 • Or ∂ 2 X ( x ) ∂ 2 Y ( y ) ∂ 2 Z ( z ) 1 1 1 + + = 0 X ( x ) ∂ x 2 Y ( y ) ∂ y 2 Z ( z ) ∂ z 2

Solving partial differential equations: separation of variables ∂ 2 X ( x ) ∂ 2 Y ( y ) ∂ 2 Z ( z ) 1 1 1 + + = 0 X ( x ) ∂ x 2 Y ( y ) ∂ y 2 Z ( z ) ∂ z 2 • Or ∂ 2 X ( x ) ∂ 2 Y ( y ) ∂ 2 Z ( z ) 1 1 1 + = − X ( x ) ∂ x 2 Y ( y ) ∂ y 2 Z ( z ) ∂ z 2 • Each side must be constant ∂ 2 Z ( z ) ∂ 2 X ( x ) ∂ 2 Y ( y ) 1 1 1 = − m + = m Z ( z ) ∂ z 2 X ( x ) ∂ x 2 Y ( y ) ∂ y 2

Solving partial differential equations: separation of variables ∂ 2 X ( x ) ∂ 2 Y ( y ) 1 1 + = m X ( x ) ∂ x 2 Y ( y ) ∂ y 2 • Again ∂ 2 X ( x ) ∂ 2 Y ( y ) 1 1 = m − X ( x ) ∂ x 2 Y ( y ) ∂ y 2 • Each side must be constant ∂ 2 X ( x ) ∂ 2 Y ( y ) 1 1 = l = m − l X ( x ) ∂ x 2 Y ( y ) ∂ y 2

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=m 0 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 —> A in = 0 —> inside ɸ constant • Boundary at infinity: set to zero —> B out = 0 -1 (1/cosh[u 0 ]) • Boundary at the shell u=u 0 : B in = A out sin

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 u 0

Potential of an oblate shell