1 : Preliminaries | 6 : Dynamics I | 11 : Star Formation | 16 : Cosmology |
2 : Morphology | 7 : Ellipticals | 12 : Interactions | 17 : Structure Growth |
3 : Surveys | 8 : Dynamics II | 13 : Groups & Clusters | 18 : Galaxy Formation |
4 : Lum. Functions | 9 : Gas & Dust | 14 : Nuclei & BHs | 19 : Reionization & IGM |
5 : Spirals | 10 : Populations | 15 : AGNs & Quasars | 20 : Dark Matter |
By the end we will have covered almost all of B&T while omitting much of the detail.
For the radial and tangential velocities, we find:
(6.1a) (6.1b) |
Where Oort's constants A and B are given by (R_{0} = solar radius):
(6.2a) (6.2b) |
Oort's A expresses local shear, while
Oort's B expresses local vorticity, ie local rotation: _{loc} =
× V (curl V)
Notice their dimensions are velocity gradients which are also frequencies
E.g. using psm units: A = 0.0148 km/s/pc 0.014 Myr^{-1} = 14.8 Gyr^{-1}
(6.3a) (6.3b) |
The first of these confirms that the rotation curve is fairly flat near the sun (gently rising).
The second yields an orbital period for the sun:
P(R_{0}) = 2 / (R_{0}) = 2 / 27.2 Gyr^{-1} = 0.23 Gyr = 230 Myr
And when coupled with an estimate for the galactocentric distance, R_{0}, yields an orbital velocity:
V_{c}(R_{0}) = 218 (R_{0} / 8 kpc) km/s
Which agrees fairly well with radio VLBI measurements of from proper motion of Sgr A^{*}
V_{c}(R_{0}) = 241 (R_{0} / 8 kpc) km/s
the cycle repeats, and we have a small retrograde epicycle
(6.4) |
(6.5) |
(6.6) |
consider small motion above and below the plane,
we simply expand the z-force linearly for small z:
(6.7) |
(6.8) |
(6.9) |
For non-circular orbits, the radial acceleration is given by (centrifugal - gravity):
(6.10) |
(6.11) |
Where the effective potential,
_{eff},
allows us to describe the radial motion in 1-D form
[image]
Typically,
_{eff}
has a minimum, rising steeply at small R and slowly at large
R
This inner steep term imposes an angular momentum (or centrifugal) barrier
At the minimum in
_{eff},
we recover the circular guiding orbit of radius R_{g}
(6.12) |
(6.13) |
This gives SHM about the guiding radius
(6.14) |
with frequency , where
(6.15) |
(6.16) |
Integration gives :
(6.17) |
(6.18) |
the oscillation of frequency
is the same as
in x, but out of phase
by 90°
(6.19) |
Some properties of this motion are:
(6.20) |
(6.21) |
Star (and gas) orbits are modified by the spiral perturbation
Their new orbits define a new surface density and associated potential.
Look for Quasi-Stationary Steady State solution (QSSS).
There are two other obvious sources of density waves, both are m = 2.
Instabilities can arise from a competition between:
Toomre (1964) found the conditions for instability: Q < 1 where Q
/
(3 G )
Where is the stellar velocity dispersion and is the local surface
density
(6.22) |
[A similar relation for gravitational stability for a gas disk is: Q V_{s} / 3 G > 1 ]
Epicyclic motion approximately follows the arm
Long perturbation duration so epicycle amplified
The emerging trailing pattern is strongly amplified
Swing amplification with feedback is probably very important in maintaining strong sprial structure.