Introduction to potential theory – at black board Potentials of simple spherical systems

1. Introduction to potential theory – at black board Potentials of simple spherical systems Point mass- keplerian potential Homogeneous sphere ρ = constant and M(r)=(4/3)πr3ρ With radial size a for r < a for r > a then 3

2. Isochrone potential – model a galaxy as a constant density at the center with density decreasing at larger radii. One potential with these properties: where b is characteristic radius that defines how the density decreases with r Density pair given in BT (2-34) and yields at center and at r >>b Modified Hubble profile – derived from SBs for ellipticals where a is core radius and j is luminosity density

3. Power-law density profile – many galaxies have surface brightness profiles that approximate a power-law over large radii If we can compute M(r) and Vc(r) If α = 2, this is an isothermal sphere (density goes as 1/r2)  Can be used to approximate galaxies with flat rotation curves; need outer cut-off to obtain finite mass

4. Plummer Sphere – simple model for round galaxies/clusters This potential “softens” force between particles in N-body simulations by avoiding the singularity of the Newtonian potential. The density profile has finite core density but falls as r-5 at large r (too steep for most galaxies). Jaffe and Hernquist profiles Both decline as r-4 at large radii which works well with galaxy models produced from violent relaxation (i.e. stellar systems relax quickly from initial state to quasi-equilibrium). Hernquist has gentle power-law cusp at small r while Jaffe has steeper cusp. Potential density

5. Density distributions for various simple spherical potentials

6. Navarro, Frenk and White (NFW) profile Good fit to dark matter haloes formed in simulations Problem – mass diverges logarithmically with r  must be cut off at large r Potentials for Flattened Models: Axisymmetric potential Kuzmin Disk (cylindrical coordinates) At points with z<0, Φk is identical with the potential of a point mass M at (R,z) = (0,a) and when z>0, Φk is the same as the potential generated by a point mass at (0,-a). Everywhere except on plane z=0

7. Use divergence theorem to find the surface density generated by Kuzmin potential Kuzmin (1956) or Toomre model 1 (1962) Miyamoto & Nagai (1975) introduced a combination Plummer sphere/Kuzmin disk model where b is aP in previous Plummer notation a=0  Plummer sphere b=0  Kuzmin disk b/a ~ 0.2 similar to disk galaxies

8. Stellar Orbits • For a star moving through a galaxy, assume its motion does not change the overall potential • If the galaxy is not collapsing, colliding, etc., assume potential does not change with time • Then, as a star moves with velocity v, the potential at its location changes as • Recall (grad of potential is force on star) • Then, • Energy along orbit remains constant (KE is only + and PE goes to 0 at large x) • Star escapes galaxy if E > 0 • Circular velocity angular velocity

9. In a cluster of stars, motions of the stars can cause the potential to change with time. The energy of each individual star is no longer conserved, only the total for the cluster as a whole. cluster KE cluster PE Stars in a cluster can change their KE and PE as long as the sum remains constant. As they move further apart, PE increases and their speeds must drop so that the KE can decrease.

10. The virial theorem tells how, on average, KE and PE are in balance Begin with Newton’s law of gravity and add an external force F Take the scalar product with xα and sum over all stars to get… VT is tool for finding masses of star clusters and galaxies where the orbits are not necessarily circular. For system in steady-state (not colliding, etc), use VT to estimate mass Assume average motions are isotropic <v2> ≈ 3σr2 KE ≈ (3σr2/2) (M/L) Ltot Get PE by M = Ltot (M/L) then use galaxy SB to find volume density of stars.

11. Orbits in Spherical Potentials – terms to know In n space dimensions, some orbits can be decomposed in n independent periodic motions – regular orbits (winding paths on a n-dimensional torus) Constants of Motion – functions of phase-space coordinates and time which are constant along the orbit C (x,v,t) = const where v = dx/dt In phase-space of 2n dimensions, there are always 2n independent constants of motion. We will see in spherical potentials, there are 4 constants of motion (2 dimensions) relating to the 4 equations of motion. Example: ϕ = Ωt + ϕ0 C = t – ϕ/Ω for a circular orbit where ϕ is the only dimension Integrals of Motion – functions of phase-space coordinates alone that are constant along any orbit I (x,v) = const Regular orbits have n isolating integrals and define a surface of 2n-1 dimensions Example: E(x,v) = ½ v2 + PE  conservation of energy along an orbit

12. Orbits in Spherical Potentials – at blackboard Each integral of motion defines a surface in 3-d space (R, VR, Vϕ) Constant E surface revolves around R-axis Constant L surface is hyperbolas in the R, Vϕ plane Intersection is closed curve and the orbit travels around this curve Vϕ VR R *note that both L and J are used to denote angular momentum The integrals of motion combine (see BT 3.1 for treatment) to produce a differential equation where u = 1/R

13. Solutions to this equation have 2 forms: bound = orbits oscillate between finite limits in R unbound = R  ∞ or u  0 Each bound orbit is associated with a periodic solution to this equation. Star in this orbit also has a periodic azimuthal motion as it orbits potential center. Relationship between azimuthal and radial periods is found to be: is usually not a rational number so orbit is not closed in most spherical potentials • star never returns to starting point in phase-space • typical orbit is a rosette and eventually passes every point in annulus between pericenter and apocenter

14. Two special potentials where all bound orbits are closed • Keplerian potential – point mass • - radial and azimuthal periods are equal • - all stars advance in azimuth by between successive pericenters • 2) Harmonic potential – homogeneous sphere • - radial period is ½ azimuthal period • - stars advance in azimuth by between successive pericenters • Real galaxies are somewhere between the two, so most orbits are rosettes advancing by •  Stars oscillate from apocenter to pericenter and back in a shorter time than is required for one complete azimuthal cycle about center

15. Orbits in Axisymmetric Potentials – at blackboard Φeff = ½ Vo2 ln (R2 + z2/q2) + Lz2/(2R2) Φ(R,z) • q= axial ratio • Resembles Φ of star in oblate spheroid with constant Vc = Vo • Φeff rises steeply toward z-axis

16. If only E and Lz constrain motion of star on R,z plane, star should travel everywhere within closed contour of constant Φeff • But, stars launched with different initial conditions with same Φeff follow distinct orbits • Implies 3rd isolating integral of motion – no analytically form

17. Nearly Circular Orbits (in axisymmetric potentials) – epicyclic approximation – at blackboard

18. Orbits in Non-Axisymmetric Potentials • Produce a richer variety of orbits – Φ = Φ (x,y,z) cartesian coordiates • Only 1 classical integral of motion – E = ½ v2 + Φ • though other integrals of motion may exist for certain potential which cannot be represented in analytical form • Orbits in non-axisymmetric potential can be grouped into Orbit Families. Examples can be found in two types of NAPs. • Separable Potentials • All orbits are regular (i.e. the orbits can be decomposed into 2 or 3 independent period motions (in 2 or 3-d) • All integrals of motion can be written analytically • These are mathematically special and therefore not likely to describe real galaxies  . However, numerical simulations for NA galaxy models with central cores have many similarities with separable potentials. • Distinct families are associated with a set of close stable orbits. In 2-d: • Oscillates back and forth along major axis (box orbits) • Loops around the center (loop orbits)

19. 2-D orbits in non-axisymmetric potential • For larger R > Rc, orbits are mostly loop orbits • initial tangential velocity of star determines width of elliptical annulus • similar to way in which width of annulus in AP varies with Lz • For small R<<Rc, orbits become box orbits • potential approximates that of homogeneous sphere • orbits are like harmonic oscillator

20. Triaxial potentials with cores have orbit families like those in separable potentials. In 3-d (triaxial potential), there are four families of orbits: box orbit: move along longest (major) axis, parent of family short axis tube orbit: loop around minor axis (resemble annular orbit of axisymmetric potential Intermediate and short axis orbits are unstable! inner long-axis tube orbit: loop around major axis outer long-axis tube orbit: loop around major axis Intermediate axis loop orbits are unstable!

21. Scale Free Potentials • All properties have either a power-law or logarithmic dependence on radius (i.e. ρ ~ r-2) • These density distributions are similar to central regions of E’s and halos of galaxies in general • If density falls as r-2 or faster, box orbits are replaced by boxlets • box orbits about minor-axis arising from resonance between motion in x and y directions (Miralda-Escude & Schwarzchild 1989) • Some irregular orbits exist as well (i.e. stochastic motions which wander anywhere permitted by conservation of energy). Stellar Dynamical Systems – at blackboard Collisionless Dynamics – at blackboard