AST 1420 Galactic Structure and Dynamics Galaxies are approx. - - PowerPoint PPT Presentation

AST 1420 Galactic Structure and Dynamics

Galaxies are approx. collisions systems

Galaxies as a collection of point masses

• Galaxies like the Milky Way are made up of ~1011

stars

• Even lower-mass galaxies are made up of >~ 107

stars and typical stellar clusters have 103-106 stars

• Do we need to compute the gravitational force by

combining GM/r^2 from all 1011 stars (+DM) to study galaxy dynamics?

Galaxies as a collection of point masses

• Basic answer is no:
• gravitational force drops as 1/r2
• For ~constant density, number of stars at distance r in shell

with width dr is r2 x dr

• Total force from this shell: dr
• Many more shells at large r than small r, and force from each
• f those is combination of many stars
• Gravitational force can therefore be approximated as a smooth field

Galaxies as collisionless systems

• Saying that the gravitational force is smooth is the

same as saying that collisions don’t matter much to the orbits of stars

• To more quantitively determine whether collisions

matter, we can compute the time necessary for close encounters to change the velocity by order unity

• Approximate treatment of galaxies allows a simple

estimate to be made (see notes):

Galaxies as collisionless systems

• Therefore, collisions are only important on

timescales >> the age of the Universe

• We can therefore usefully treat galaxies as smooth

mass distributions

Dense star clusters are collisional systems

• Therefore, collisions in dense star clusters are

important on timescales ~ the age of the Universe

• Dynamics of dense star clusters is much more

complicated!

• Dense star clusters have crossing times of ~1 Myr

and ~105 stars

Spherical mass distributions

The ‘spherical cow’ treatment of galaxies

Many of the galaxies that we are interested in look like this:

• Spherical approximation still useful:
• To ‘zero-th order’: radial behavior of a mass

distribution most important for its dynamics

• Dark-matter distribution ~ spherical, so much

research on DM dynamics uses spherical potentials and orbits in spherical potentials

• Far easier to work with spherical potentials (force-law,
• rbits, equilibria) than at the next order of approximation

The ‘spherical cow’ treatment of galaxies

Gravitational potential theory

Gravitational force

• Consider gravitational force: the force from mass M
• n a body with mass m the force is F = -GMm/r2

along direction connecting the two; G gravitational constant

• If M is distributed in space in parcels dM(x)

and

Newton’s second law

• F = m a
• For gravitational force: F = m g
• a = g!
• Gravitational field —> acceleration due to gravity
• So typically do not really distinguish between force,

field, and acceleration

Gravitational potential

• Thus, we can write the gravitational field as the

gradient of a potential

• This simplifies the description: 3D vector field g

reduced to 3D scalar field ɸ

• This further means that the gravitational force is

conservative: work done in moving from x1 to x2 does not depend on path

Gravitational potential

ɸ(x) ???

The Poisson equation

• This is the Poisson equation
• Relates the scalar density to the scalar potential; often easiest

way to solve for the gravitational force from a given density

• Clearly linear: ⍴1+⍴2 —> ɸ1+ɸ2
• One of the fundamental equations of galactic dynamics!
• Bunch of steps….

Potential of a spherical mass distribution

• Problem: given density ⍴(r), what is ɸ(r)?
• Solve Poisson equation?
• Two theorems by Newton (!) significantly simplify

finding ɸ(r)

Potential of a spherical mass distribution Potential of a spherical mass distribution

Potential of a spherical mass distribution Potential of a spherical mass distribution

• Newton’s first theorem —-> potential within the shell ==

constant (force = d ɸ / d r = 0)

• Thus, can evaluate potential at any point within the shell
• Easiest at r=0
• Each point along the shell contributes -G Ms / [4πR^2] / R
• Integrate over entire surface: 4πR^2 x -G Ms / [4πR^2] / R

= -GMs/R

Potential of a spherical shell with mass Ms and radius R

• Newton’s second theorem: force the same as if all

mass were concentrated in a point —> potential the same

• Potential of a point-mass is ɸ = -GM/r

Potential of a spherical shell with mass Ms and radius R

Potential of a spherical shell with mass Ms and radius R

ɸ r

ɸ r

Potential of any spherical mass distribution

Potential of any spherical mass distribution

• ⍴(r) can be composed into shells with mass

4πr2⍴(r)dr

• Potential of each shell as in previous slide

• Force must be radial, because potential only depends on r
• Newton’s first law: no force from shells outside current radius r
• Newton’s second law: each shell Ms inside r gives -GMs/r2
• Total force: -GM(<r)/r2
• Mass outside current r has no influence on the acceleration
• For general mass distributions: effect of mass outside of r will

typically be quite subtle (monopole has no effect)

Force of any spherical mass distribution

Alternative expression for potential of spherical mass

Circular velocity

Circular velocity

• The circular velocity is the velocity of a body on a

circular orbit

• Circular orbit acceleration is centripetal: vc2/r
• Equal to radial force (both point inwards):



 vc2/r = GM(<r) / r2

Circular velocity measures enclosed mass (for spherical potentials)

• Galaxies are observed to have vc ~ constant
• E.g., in the Milky Way near the Sun: r ~ 8 kpc, vc

= 220 km/s

Dynamical time

• Typical dynamical time: period of circular orbit at

radius r

• tdyn = 2πr/vc

• Very important relation!
• Given estimate of average density —> typical
• rbital time

Dynamical time

Dynamical time examples

Dynamical time examples

Energy

• Since the gravitational force is conservative, we

can define an energy E

• In a static potential, energy is conserved

Escape velocity

• Potentials that approach a finite value at r=infinity allow unbound
• rbits: E >= ɸ(∞)
• Such orbits can escape the potential (leave and never come back)
• Boundary: E = ɸ(∞)
• r

Escape velocity

• If we could measure the escape velocity at a point r,

it would directly tell us about the potential ɸ(r) at r

• From the expression of the potential we see that this

is very powerful

Escape velocity near the Sun

• We will discuss later how we can measure the

escape velocity near the Sun

• Escape velocity is ~550 km/s
• Compare to circular speed = 220 km/s
• If no mass outside:

• 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

Escape velocity near the Sun

• From

Escape velocity near the Sun

Some examples of spherical potentials

Point mass

Homogeneous density sphere

Plummer sphere

→

Isochrone potential

→

Isochrone potential

Rotation curves of these examples

Power-law models

always a convenient model in astrophysics!

Power-law models

always a convenient model in astrophysics!

Two-Power-law models

if one power-law doesn’t fit, try a broken power-law!

• Two important parameter settings:
• Hernquist: alpha=1, beta=4 —> elliptical galaxies, bulges, DM

halos

• NFW: alpha=1, beta=3 —> dark-matter halos

Two-Power-law models

NFW profile

• “Standard” model for dark-matter halos: resulting DM profile in

cosmological simulations of structure formation, origin not particularly well understood

• Parameterized in many different ways:
• Mass, concentration: mass to ‘virial radius’, concentration =

(virial radius)/a

• Vmax, rmax: peak of the rotation curve and radius of the peak
• ⍴0 and a…

Hernquist profile

• Simple model for bulges, elliptical galaxies, and

DM halos

• More tractable than NFW: has finite mass and

simple form of potential

• In the future, dark-matter halos will tend to

Hernquist profiles

Hernquist and NFW: rotation curves

Hernquist and NFW: rotation curves

Elements of classical mechanics

Classical mechanics

• Orbits in gravitational potential is a subset of the

wider area of classical mechanics

• Not covered in the notes, but look at Appendix D of

Binney & Tremaine (2008) for brief overview or any book on classical mechanics

Why classical mechanics?

• Could just write down the equations of motion 


F = m a = m d2x / dt2
 in whatever coordinate system you want, but tools of classical mechanics make this much easier

• Similarly, many other aspects of gravitational dynamics are simpler to

understand in the framework of classical mechanics:

• Theory of evolution of the distribution functions of many bodies
• Dynamical equilibria of galaxies
• Numerical orbit integration
• Slow evolution of gravitational systems
• …

Hamilton’s principle and Lagrange’s equations

• For a body in a conservative force field we can

introduce a kinetic energy K [=|v|2/2] and a potential energy V [=ɸ(x)]

• Lagrangian: L = K-V = |v|2/2-ɸ(x)
• Hamilton’s principle: motion of body such that ∫dt L is

minimized

• Calculus of variations allows derivation of Lagrange’s

equation

Lagrange’s equation

• The powerful aspect of Lagrange’s equation is that

it holds for arbitrary coordinates q and qdot = dq/ dt

• This can make it much easier to derive the

equations of motion in non-cartesian coordinate frames

Example: polar coordinates

Hamiltonian dynamics

• But there’s more!
• For cartesian coordinates x we have momenta p =

mv (where m=1 for momenta/unit mass)

• From the Lagrangian, we can define a set of

generalized momenta p for coordinates q

• The Hamiltonian is defined as

Hamiltonian dynamics

• From taking the total derivative d H and using

Lagrange’s equation we can derive Hamilton’s equations

• The equations allow one to solve for the evolution of

phase-space coordinates (q,p)

• Let’s look at a time-independent potential

Hamiltonian dynamics

• This is simply equal to the energy, which is conserved
• The Hamiltonian framework makes it easy to work with phase-space

coordinates in general coordinate systems

• Furthermore, it allows a wide range of canonical transformations of w =

(q,p) —> (q’,p’) that can significantly simplify the dynamics of a gravitational system

• One of the most important properties of canonical transformations is that

they have a Jacobian J with |J| = 1 such that probability densities are the same in any canonical coordinates system p(q,p) = p(q’,p’) for any p(.,.)

Orbits in spherical potentials

Equations of motion in a spherical potential

• Only non-zero component of the force in spherical

coordinates is the radial component and Newton’s second law becomes

• This implies that the total angular momentum L = r

x dot is conserved

The orbital plane

• Because the angular momentum vector is

conserved, position r and velocity rdot are always perpendicular to constant L —> motion is confined to a plane perpendicular to the angular momentum vector

• Thus, we can focus on the motion within the orbital

plane

Motion in the orbital plane

• The equations of motion in the orbital plane are

simply those that we derived in polar coordinates earlier

• The second of these is just another manifestation of

the conservation of angular momentum: the magnitude

• f the angular momentum must be conserved

• The first equation can be written in terms of an

effective potential

Motion in the orbital plane

• With energy

Effective potential

E angular momentum barrier

Pericenter, apocenter, eccentricity

• Orbit with angular momentum L oscillates radially

between rp and ra, pericenter and apocenter

• Can define measure of how circular an orbit is, the
• rbital eccentricity

Radial and azimuthal period

• Radial oscillation has a period Tr: the radial period
• In one radial period go through the following range

in azimuth

• And the azimuthal period is

Now let’s look at some actual orbits!

Orbits in the homogeneous sphere

• Potential is that of a harmonic oscillator
• and the equations of motion are therefore
• with solution
• This is the equation of an ellipse with the center at the
• rigin

• Energy and angular momentum

Orbits in the homogeneous sphere

• Pericenter and apocenter
• eccentricity = (b-a)/((b+a)
• Simplify to

Orbits in the homogeneous sphere

• Radial and azimuthal period:

Orbits in the homogeneous sphere

Orbits in the homogeneous sphere

Orbits in the homogeneous sphere

• Convert time derivatives to azimuthal derivatives in the EOM

Keplerian orbits

• and then convert the EOM to an equation for u = 1/r
• For the Kepler potential

• is the equation of a forced harmonic oscillator with solution

Keplerian orbits

• or
• This is the equation of an ellipse with the focus at the
• rigin

Keplerian orbits

• Semi-major axis and eccentricity
• Pericenter and apocenter
• Energy

• Radial and azimuthal period are equal

Keplerian orbits

Keplerian orbits

Keplerian orbits

Keplerian orbits

Orbits in the homogeneous sphere vs. in the Kepler potential

• Homogeneous sphere to Kepler potential spans the range
• f plausible spherical potentials
• Orbits are ellipses in both!
• Homogeneous sphere: radial period = azimuthal period / 2
• Kepler potential: radial period = azimuthal period
• Thus, for any spherical potential the radial period is

somewhere between half and once the azimuthal period —> stars oscillate radially more rapidly than azimuthally

• Radial period only depends on E, not on L

Orbits in the isochrone potential

• Azimuthal range in one radial period :

Orbits in the isochrone potential

Next week

• General theory of dynamical equilibrium
• Equilibria of spherical systems