Transcript No Slide Title

The Local Group galaxies: M31, M32, M33, and others.
Dwarfs in our neighborhood
Galaxy groups within 80/h Mpc from us
Local Group
Ch 4. Our backyard: The local group
!
Local Group =
Only 3 spirals
Only 1 elliptical (!)
Lots of dwarf, irregular
galaxies
One more view of the Local Group...
M31 = Andromeda galaxy (Sb)
M32
NGC 205
100 Mpixel CCD camera CFHT12K
on CFHT = Canada-France Hawaii telescope
LMC = Large Magellanic Cloud, a neighbor bound to the
Milky Way
(SBm)
Rotation
speed
~80 km/s
SMC = Small
Magellanic Cloud
(Irr)
trajectory similar to LMC,
bound to it
Magellanic Stream is gas
shed by Magellanic Clouds
No rotation
NGC 6822
(Irr)
Fornax dwarf spheroidal galaxy
(dSph)
Foreground
MW star
4.1.4. Life in orbit: the tidal limit
This is a standard 3-Body problem, the larger
mass m1 is a big galaxy, and m2 a small dwarf
galaxy or a globular cluster.
The third body, a massless test particle, is a star in
the companion (smaller) system.
Circular Restricted 3-Body Problem (R3B)
L4
L3
Joseph-Louis Lagrange (1736-1813)
[born: Giuseppe Lodovico Lagrangia]
L1
L2
L5
“Restricted” because the gravity of particle moving around the
two massive bodies is neglected (so it’s a 2-Body problem plus 1 massless
particle, not shown in the figure.)
Furthermore, a circular motion of two massive bodies is assumed.
General 3-body problem has no known closed-form (analytical) solution.
NOTES:
The derivation of energy (Jacobi) integral in R3B does not differ
significantly from the analogous derivation of energy
conservation law in the inertial frame, e.g., we also form the
dot product of the equations of motion with velocity and
convert the l.h.s. to full time derivative of specific kinetic energy.
On the r.h.s., however, we now have two additional accelerations
(Coriolis and centrifugal terms) due to frame rotation (non-inertial,
accelerated frame). However, the dot product of velocity and the
Coriolis term, itself a vector perpendicular to velocity, vanishes.
The centrifugal term can be written as a gradient of a
‘centrifugal potential’ -(1/2)n^2 r^2, which added to the usual sum
of -1/r gravitational potentials of two bodies, forms an effective
potential Phi_eff. Notice that, for historical reasons, the effective
R3B potential is defined as positive, that is, Phi_eff is the sum of
two +1/r terms and +(n^2/2)r^2
n
Effective potential in R3B
mass ratio = 0.2
The effective potential of R3B is defined as negative of the usual Jacobi
energy integral. The gravitational potential wells around the two bodies thus
appear as chimneys.
Lagrange points L1…L5 are equilibrium points in the circular
R3B problem, which is formulated in the frame corotating with
the binary system. Acceleration and velocity both equal 0 there.
They are found at zero-gradient points of the effective potential
of R3B. Two of them are triangular points (extrema of potential).
Three co-linear Lagrange points are saddle points of potential.
Jacobi integral and the topology of Zero Velocity Curves in R3B
  m1 /(m1  m2 )
rL = Roche lobe radius
+ Lagrange points
Sequence of allowed regions of motion (hatched) for particles
starting with different C values (essentially, Jacobi constant ~
energy in corotating frame)
High C (e.g., particle
starts close to one of
the massive bodies)
Highest C
Medium C
Low C (for instance,
due to high init.
velocity)
Notice a curious fact:
regions near L4 & L5
are forbidden. These
are potential maxima
(taking a physical, negative
gravity potential sign)
Roche lobe radius depends weakly on R3B mass parameter
  m1 /(m1  m2 )
= 0.1
  m1 /(m1  m2 ) = 0.01
Computation of Roche lobe radius from R3B equations
of motion (   x / a , a = semi-major axis of the binary)
L
Roche lobe radius depends weakly on R3B mass parameter
 1/ 3
rL  ( 3 ) a
  m1 /(m1  m2 )
= 0.1
  m1 /(m1  m2 ) = 0.01
m2/M = 0.01 (Earth ~Moon) r_L = 0.15 a
m2/M = 0.003 (Sun- 3xJupiter) r_L = 0.10 a
m2/M = 0.001 (Sun-Jupiter) r_L = 0.07 a
m2/M = 0.000003 (Sun-Earth) r_L = 0.01 a
Our textbook calls Roche lobe radius (Hill radius) the
“Jacobi radius” rJ , to indicate that in galactic
dynamics, the potentials involved are rarely those of point
masses (for instance, potential and rotation curve of our Galaxy
are clearly different).
Thus, the problem is only approximately a R3B.
Indeed, number ‘3’ by which the mass ratio is divided
in the formula for rL gets replaced by ‘2’, if instead of point-like
mass distribution (a black hole in the center?) the potential of
a galaxy is modeled as a singular isothermal sphere or a
disk potential that produces a flat rotation curve (see problem
4.4 in Sparke & Gallagher.)
Please read about the Local Group from Ch.4., omitting
the chemical evolution section.