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 ~10 11 stars • Even lower-mass galaxies are made up of >~ 10 7 stars and typical stellar clusters have 10 3 -10 6 stars • Do we need to compute the gravitational force by combining GM/r^2 from all 10 11 stars (+DM) to study galaxy dynamics?
Galaxies as a collection of point masses • Basic answer is no: • gravitational force drops as 1/r 2 • For ~constant density, number of stars at distance r in shell with width dr is r 2 x dr • Total force from this shell: dr • Many more shells at large r than small r, and force from each of 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 • Dense star clusters have crossing times of ~1 Myr and ~10 5 stars • Therefore, collisions in dense star clusters are important on timescales ~ the age of the Universe • Dynamics of dense star clusters is much more complicated!
Spherical mass distributions
The ‘spherical cow’ treatment of galaxies Many of the galaxies that we are interested in look like this:
The ‘spherical cow’ treatment of galaxies • 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, orbits, equilibria) than at the next order of approximation
Gravitational potential theory
Gravitational force • Consider gravitational force: the force from mass M on a body with mass m the force is F = -GMm/r 2 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
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 x 1 to x 2 does not depend on path
ɸ ( x ) ???
The Poisson equation • Bunch of steps…. • 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 —> ɸ 1 + ɸ 2 • One of the fundamental equations of galactic dynamics!
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 Potential of a spherical mass distribution distribution
Potential of a spherical mass Potential of a spherical mass distribution distribution
Potential of a spherical shell with mass M s and radius R • 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 M s / [4 π R^2] / R • Integrate over entire surface: 4 π R^2 x -G M s / [4 π R^2] / R = -GM s /R
Potential of a spherical shell with mass M s 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 M s and radius R ɸ r
Potential of any spherical mass distribution ɸ r
Potential of any spherical mass distribution • ⍴ (r) can be composed into shells with mass 4 π r 2 ⍴ (r)dr • Potential of each shell as in previous slide
Force of any spherical mass distribution • 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 M s inside r gives -GM s /r 2 • Total force: -GM(<r)/r 2 • 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)
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: v c2 /r • Equal to radial force (both point inwards): v c2 /r = GM(<r) / r 2
Circular velocity measures enclosed mass (for spherical potentials) • Galaxies are observed to have v c ~ constant • E.g., in the Milky Way near the Sun: r ~ 8 kpc, v c = 220 km/s
Dynamical time • Typical dynamical time: period of circular orbit at radius r • t dyn = 2 π r/v c
Dynamical time • Very important relation! • Given estimate of average density —> typical orbital 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 orbits: E >= ɸ ( ∞ ) • Such orbits can escape the potential (leave and never come back) • Boundary: E = ɸ ( ∞ ) or
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:
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
Escape velocity near the Sun • From
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
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