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-vesc| < 200 km/s
• Escape velocity manifests itself as a cut-off in the velocity

distribution: no stars with velocities > vesc

• Simple estimator vesc = 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 vesc 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)

• 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/r2 out to 100 kpc, difference in mass

between 100 and 8 kpc leads to potential difference

Week 2: 
 Escape velocity near the Sun

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 = MMW

MM31 / M around point mass with mass M = MMW + MM31

• 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 Tr, which we could get by solving

for the entire orbit

• But we can estimate Tr as Tr = tH -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 Tr and underestimates M
• Lower limit on Tr = tH
• This gives M = 5.5 x 1012 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’ Tr = 16.6 Gyr —> 4.6 x 1012 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/r3.5 (e.g., Bell et al. 2008)
• Velocity dispersion
• Anisotropy β: unknown

Xue et al. (2008)

Inferred “rotation curve”

Xue et al. (2008)

Xue et al. (2008)

Milky Way mass from halo “rotation curve”

Masses of dwarf spheroidal galaxies

Dwarf spheroidal galaxies

• Small galaxies with stellar masses ~106 to 108 Msun
• 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?

Walker et al. (2009)

Dwarf spheroidal galaxies: kinematics

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

f(x) = Z ∞

x

dt g(t) √t − x g(t) = − 1 π Z ∞

t

dx 1 √x − t df dx

• We will typically need this for α=1/2

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 σlos = σr
• 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

σlos and compare to data

Walker et al. (2009)

Dwarf spheroidal galaxies: masses

M/L ~ 10 — 100!

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 σlos ~ constant

• First shown by Wolf et al. (2010)
• Start with
• Can write this as
• Abel transform!

Breaking the mass- anisotropy degeneracy

}

• bservable —> independent of β

independent of β

}

Breaking the mass- anisotropy degeneracy

• Derivative wrt ln r:
• Use the Jeans equation

}

where = 0

}

=0 —> independent of β

Breaking the mass- anisotropy degeneracy

• For σlos ~ 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 ~ r1/2

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 σlos ~ constant:

Wolf mass estimator

• Use Jeans:
• +
• Gives the Wolf mass estimator
• in terms of r1/2 or Re (=2D R1/2)

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

van Dokkum et al. (2015)

Coma cluster

van Dokkum et al. (2015)

Koda et al. (2015)

Beasley et al. (2016)

Masses of UDGs from the Wolf estimator

• Globular clusters in VCC 1287:
• Don’t know about σlos(r), but let’s apply the Wolf mass estimator

(large) dwarf galaxy
 anomalously high M/L

Dragonfly 44

van Dokkum et al. (2016)

Mass of Dragonfly 44

• From Keck spectra can measure velocity dispersion:

σlos(r) ~ constant

Masses of UDGs

van Dokkum et al. (2016)

• Dragonfly 44: M/L ~ 50 within R<~5 kpc!
• Comparison Milky Way: M/L ~ few within the same radius
• Total mass of Dragonfly 44 (extrapolation): ~1012 Msun ~

MMW

• Dragonfly 44 ~ MW, but only 1% of stars —> failed MW?
• VCC 1287 —> dwarf galaxy, but high M/L —> failed

dwarf?

• Unclear why star formation is so low in these galaxies

Masses of UDGs

Inner dark matter profile of dwarf spheroidal galaxies

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

Dark matter in the smallest galaxies

• Cosmological simulations of formation of dark matter halos —> NFW

profile with inner cusp ρ(r) ~ 1/r

• Because the smallest galaxies


are the most dark-matter 
 dominated, easiest to measure
 the dark matter profile there 
 (M/L ~ 10s vs few for MW)

• Results from dark matter being 


collisionless —> if DM has 
 (self-)interactions cusp can be 
 destroyed and become core 
 ρ(r) ~ constant

• Feedback from supernovae can also 


turn cusp into core 
 (e.g., Pontzen & Governato 2012)

Ebert et al. (2016)

Dark matter in the smallest galaxies

• Because Jeans analyses

are so dependent on the anisotropy, cannot distinguish between core and cusp in Milky Way dwarf spheroidal galaxies

Walker & Penarrubia (2011)

• Classical dwarf

spheroidals often have multiple populations

• Multiple episodes of star

formation —> populations with different ages and metallicities

• These populations have

different density distributions

Multiple populations in dwarf spheroidal galaxies

Tolstoy et al. (2004); Sculptor

Jeans modeling with multiple populations

• Wolf estimator: robust mass at r1/2,

independent of anisotropy

• Different populations have different density

profiles —> different r1/2

• Applying Wolf estimator to two populations

—> mass at two different radii

• Slope of the mass: 𝛥 = d ln M / d ln r
• Cusp: M ~ r2 —> 𝛥 = 2
• Core: M ~ r3 —> 𝛥 = 3

Walker & Penarrubia (2011)

Jeans modeling with multiple populations

Cores in dwarf spheroidal galaxies

• Data strongly rule out cusp in dwarf spheroidal

galaxies

• Mass estimator found to be robust again many of

the assumptions

• Could mean that DM is not collisionless
• Or (uncertain) baryonic effects turn cusp into core
• Baryonic effects should be less for lower masses or

correlated with star-formation history —> future tests

Presentations

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