AST 1420 Galactic Structure and Dynamics Last week: equilibrium of - PowerPoint PPT Presentation

AST 1420 Galactic Structure and Dynamics

Last week: equilibrium of dynamical systems • Collisionless Boltzmann equation: • Spherical Jeans equation: vel. moment of CBE • Jeans theorem: equilibrium DF = f(I), I integral of motion • Spherical distribution functions, e.g., singular isothermal sphere

Equilibrium spherical distribution functions • Work with relative potential • and relative energy • For ergodic DF, the density is

Milky Way escape velocity

Escape velocity in the solar neighborhood • Week 2: • Sensitive to total mass in the potential well

Escape velocity in the solar neighborhood • Difficult to measure: • Stars with high velocities wrt the Sun are rare • Partly because stars on these types of orbits are rare • and because stars with low binding energy spend little time near the Sun Hunt et al. (2016)

Escape velocity in the solar neighborhood • Thus, we need large data sets • So far mainly done with RAVE : RAdial Velocity Experiment • RAVE has ~450,000 stars • Only a few dozen with |v-v esc | < 200 km/s • Escape velocity manifests itself as a cut-off in the velocity distribution: no stars with velocities > v esc • Simple estimator v esc = max(v) • But can do better with modeling

Equilibrium modeling of the tail of the velocity distribution • Assumption 1: distribution is in equilibrium —> suspect because of long dynamical times, but seems to work from simulations • Assumption 2: potential is spherical —> not near the Sun, but stars near v esc spend most of their orbit >> disk, where the potential is (close to) spherical • Assumption 3: distribution is ergodic —> again suspect, because origin of these stars can easily induce anisotropy

Ergodic equilibrium modeling of the tail of the velocity distribution • Use f( ℰ ) model as discussed last week • Many possible models, but only interested in distribution near the escape velocity or ℰ ~ 0 • General form for f( ℰ ) ~ power-law near ℰ = 0

Ergodic DFs near ℰ = 0

Ergodic equilibrium modeling of the tail of the velocity distribution Leonard & Tremaine (1990); Kochanek (1996)

Observational constraints Smith et al. (2007)

Observational constraints Piffl et al. (2014)

Week 2: 
 Escape velocity near the Sun • The fact that the escape velocity is >> 300 km/s means that there must be much mass outside of the solar circle • How much? We can estimate • For a density ~ 1/r 2 out to 100 kpc, difference in mass between 100 and 8 kpc leads to potential difference

Milky Way mass from escape velocity (better model for the potential)

Local Group timing argument

The Local Group timing argument • One of the earliest indications of the presence of dark-matter halos came from the Local Group timing argument (Kahn & Woltjer 1959) • Argument is simple: Milky Way and M31 were at dr=0 at the Big Bang and launched on a radial orbit, currently during the first period (otherwise merger) • Radial orbit characterized by 3 unknowns: initial velocity, current phase, and mass of the system • 3 observables: current separation, velocity, and time (age of the Universe) • Solve!

The Local Group timing argument • Orbit is that of two point masses, similar to Earth around Sun (but mass ratio ~1) • Equivalent to Keplerian orbit of mass μ = M MW M M31 / M around point mass with mass M = M MW + M M31 • Traditional solution solves for the entire orbit, rather tedious… • Let’s use conservation of energy and angular momentum to do a simpler estimate!

The Local Group timing argument • Energy conservation: • Or • With Kepler’s third law

The Local Group timing argument • Becomes cubic equation for x = GM: • To solve this, we need T r , which we could get by solving for the entire orbit • But we can estimate Tr as T r = t H -r/v = 19.5 Gyr (r = 740 kpc, v = -125 km/s)

The Local Group timing argument

The Local Group timing argument • Previous estimate overestimates T r and underestimates M • Lower limit on T r = t H • This gives M = 5.5 x 10 12 Msun • Li & White (2008) show using Millennium simulation halos that the timing mass has a scatter of a factor of two around the true value, so our estimate is pretty good • ‘True’ T r = 16.6 Gyr —> 4.6 x 10 12 Msun

The mass of the Milky Way’s dark matter halo

Mass of the Milky Way • From escape velocity and timing argument, clearly the Milky Way has lots of dark matter • But how much and how is it distributed? • Cannot use disk stars or gas beyond ~15 kpc to measure rotation curve (see next weeks…) • But can use halo tracers: globular clusters (e.g., last week) or halo stars • If we assume dynamical equilibrium, can tackle this with the spherical Jeans equation

Mass of the Milky Way: spherical Jeans equation • From last week: • In terms of the circular velocity • Thus, can use measurements of density nu and velocity dispersion to measure the “rotation curve’’

Spherical Jeans ingredients • Density: ~ 1/r 3.5 (e.g., Bell et al. 2008) • Velocity dispersion • Anisotropy β : unknown

Xue et al. (2008)

Inferred “rotation curve” Xue et al. (2008)

Milky Way mass from halo “rotation curve” Xue et al. (2008)

Masses of dwarf spheroidal galaxies

Dwarf spheroidal galaxies • Small galaxies with stellar masses ~10 6 to 10 8 M sun • Most easily seen around the Milky Way: • Classical dwarf spheroidals: Fornax, Sculptor, … • Ultra-faints: discovered by SDSS (now DES), much lower surface brightness: Wilman 1, SEGUE-2, …

Dwarf spheroidal galaxies • Interesting laboratories for galaxy formation and dark matter: • Smallest galaxies that form —> how do stars in small galaxies, affected by reionization of the Universe • Chemical evolution: stars affected by few SNe / other processes —> see effect from rare stages of stellar evolution (e.g., r-process enhancement from neutron star mergers) • High mass-to-light ratios —> dark-matter dominated: distribution of dark matter • Good targets for dark-matter annihilation • Important to have good constraints on their masses and mass profiles —> what is the potential well?

Dwarf spheroidal galaxies: kinematics Walker et al. (2009)

Dwarf spheroidal galaxies: density • We measure the projected surface density 𝛵 (R) • For the Jeans equation, we need to know the 3D density ν (R) • Can write this in terms of ν (R) • Abel integral transform, can be inverted

Abel integral inversion • We will typically need this for α =1/2 Z ∞ Z ∞ g ( t ) g ( t ) = − 1 1 d f f ( x ) = d t d x √ t − x √ x − t d x π t x

Dwarf spheroidal galaxies: density deprojection • can therefore be inverted to

Dwarf spheroidal galaxies: kinematics deprojection • We can only measure the line-of-sight velocity • Cannot just assume that σ los = σ r • Similar deprojection formula (Binney & Mamon 1982)

Dwarf spheroidal galaxies: kinematic modeling • If we assume that the anisotropy is constant, we can integrate the Jeans equation • Thus, for a given mass M(<r) and constant anisotropy, can predict σ los and compare to data

Dwarf spheroidal galaxies: masses M/L ~ 10 — 100! Walker et al. (2009)

Breaking the mass- anisotropy degeneracy

The mass-anisotropy degeneracy • Mass-anisotropy degeneracy in the Jeans equation is a significant problem when inferring masses of stellar systems • Various groups noticed that the inferred mass is very insensitive to β around the half-light radius: Wolf et al. (2010)

Breaking the mass- anisotropy degeneracy • We can demonstrate that there is a radius at which the inferred mass from the Jeans equation does not depend on β for systems with σ los ~ constant • First shown by Wolf et al. (2010) • Start with • Can write this as • Abel transform!

Breaking the mass- anisotropy degeneracy } } observable —> independent of β independent of β

Breaking the mass- anisotropy degeneracy • Derivative wrt ln r: • Use the Jeans equation } } =0 —> independent of β where = 0

Breaking the mass- anisotropy degeneracy • For σ los ~ constant, this is dominated by • Mass independent of anisotropy where the log. slope of the density = -3! • For typical stellar profiles, happens to be that this radius ~ r 1/2

Wolf mass estimator • What is the mass at the radius where the mass is independent of β ? • Can set β = 0, because mass does not dependent on β , + take d / d ln r • For σ los ~ constant:

Wolf mass estimator • Use Jeans: • + • Gives the Wolf mass estimator • in terms of r 1/2 or R e (=2D R 1/2 )

Masses of ultra-diffuse galaxies

Ultra-diffuse galaxies • Galaxies with very low surface brightness and luminosity • Similar luminosity to dSphs, but sizes similar to MW- like galaxies • Unclear how such objects form! • Knowing their masses would help figure out how they form

Coma cluster van Dokkum et al. (2015)

van Dokkum et al. (2015)

Koda et al. (2015)

Beasley et al. (2016)