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ASTC22. Lecture L12 The Local Group

Discover the diverse galaxies in our cosmic neighborhood, including spirals, ellipticals, dwarfs, and irregulars within 80 Mpc from us. Learn about the Andromeda Galaxy (M31), the Large Magellanic Cloud (LMC), the Small Magellanic Cloud (SMC), and more. Dive into the fascinating dynamics of the Circular Restricted 3-Body Problem in galactic systems.

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ASTC22. Lecture L12 The Local Group

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  1. ASTC22. Lecture L12 The Local Group The Local Group galaxies: M31, M32, M33, and others. Dwarfs in our neighborhood

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

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

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

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

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

  7. 100 Mpixel CCD camera CFHT12K on CFHT = Canada-France Hawaii telescope

  8. 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

  9. NGC 6822 (Irr)

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

  11. 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.

  12. Circular Restricted 3-Body Problem (R3B) L4 L1 L3 L2 Joseph-Louis Lagrange (1736-1813) [born: Giuseppe Lodovico Lagrangia] 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.

  13. 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

  14. 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.

  15. 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.

  16. Jacobi integral and the topology of Zero Velocity Curves in R3B rL = Roche lobe radius + Lagrange points

  17. 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 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) Medium C

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

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

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

  21. 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.

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