1. The Local Group galaxies: M31, M32, M33, and others. Dwarfs in our neighborhood

2. Galaxy groups within 80/h Mpc from us Local Group

4. Ch 4. Our backyard: The local group !

5. Local Group = Only 3 spirals Only 1 elliptical (!) Lots of dwarf, irregular galaxies

8. One more view of the Local Group...

9. M31 = Andromeda galaxy (Sb) M32 NGC 205

10. 100 Mpixel CCD camera CFHT12K on CFHT = Canada-France Hawaii telescope

12. LMC = Large Magellanic Cloud, a neighbor bound to the Milky Way (SBm) SMC = Small Magellanic Cloud (Irr) trajectory similar to LMC, bound to it Magellanic Stream is gas shed by Magellanic Clouds Rotation speed ~80 km/s No rotation

13. NGC 6822 (Irr)

14. Fornax dwarf spheroidal galaxy (dSph) Foreground MW star

15. 4.1.4. Life in orbit: the tidal limit This is a standard 3-Body problem, the larger mass m 1 is a big galaxy, and m 2 a small dwarf galaxy or a globular cluster. The third body, a massless test particle, is a star in the companion (smaller) system.

16. Circular Restricted 3-Body Problem (R3B) L1L1 L4L4 L5L5 L3L3 L2L2 “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. Joseph-Louis Lagrange (1736-1813) [born: Giuseppe Lodovico Lagrangia]

17. 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

19. 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.

20. Lagrange points L 1 …L 5 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.

21. r L = Roche lobe radius + Lagrange points Jacobi integral and the topology of Zero Velocity Curves in R3B

22. Sequence of allowed regions of motion (hatched) for particles starting with different C values (essentially, Jacobi constant ~ energy in corotating frame) Highest C Medium C High C (e.g., particle starts close to one of the massive bodies) Low C (for instance, due to high init. velocity) Notice a curious fact: regions near L 4 & L 5 are forbidden. These are potential maxima (taking a physical, negative gravity potential sign)

23. = 0.1 = 0.01 Roche lobe radius depends weakly on R3B mass parameter

24. Computation of Roche lobe radius from R3B equations of motion (, a = semi-major axis of the binary) L

25. = 0.1 = 0.01 Roche lobe radius depends weakly on R3B mass parameter m 2 /M = 0.01 (Earth ~Moon) r_L = 0.15 a m 2 /M = 0.003 (Sun- 3xJupiter) r_L = 0.10 a m 2 /M = 0.001 (Sun-Jupiter) r_L = 0.07 a m 2 /M = 0.000003 (Sun-Earth) r_L = 0.01 a

26. Our textbook calls Roche lobe radius (Hill radius) the “Jacobi radius” r J, 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 r L 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.