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Jacobi radius in flat-Vc galaxies
Chemical evolution models
G-dwarf problem
Computation of Roche lobe in R3B (Restricted 3 Body) equations
of motion (   x / a , a = semi-major axis of the binary)
L
Point masses
only!
Jacobi radius rJ
We consider one large and one small, massive stellar systems, circling each other.
The larger one might represent the Milky Way, the small either a globular cluster,
or a dwarf galaxy merging with the Galaxy.
The derivation of tidal radius proceeds along the lines of the Roche lobe size
derivation in R3B (see Lecture 11). One difference is that force of the Galaxy is not ~1/r^2.
Three forces acting on a test particle placed at the Lagrange point L1
are summed up and equated with zero: two gravitational pulls toward the two massive
bodies, plus the centrifugal force.
Let’s denote as r_J the Jacobi or tidal radius (m2-L1 distance), and as r_0 the distance
between bodies 1 and 2; point L1 is then at r = r_0 - r_J (we count the r from body m1).
(1) force from body 1 = -G M(r_0 - r_J) /(r_0 - r_J)^2 ~ -GM(r_0)/r_0^2 *(1 + r_J/r_0)
where we took into account M(r_0-r_J) = M(r_0) *(1 - r_J/r_0) in a flat-Vc galaxy.
(2) force from body 2 = +Gm/ r_J^2
(3) centrifugal force = v_c^2 /(r0-r_J) ~ +GM(r_0)/r_0^2 *(1 - r_J/r_0)
because this force is a constant rotation speed Omega^2 times the distance r, so
it’s linearly scaling with r, thus being (1- r_J/r_0) times smaller at L1 than at r=r_0.
Dividing by the unit force (acceleration) GM(r_0)/r_0^2 we obtain, equating the
sum of the three terms to zero, and denoting x=r_J/r_0, and M_0=M(r_0), we obtain:
-(1 + x) + (m/M_0)/x^2 + (1 - x) = 0, therefore 2x = (m/M_0)/x^2, or
r_J/r_0 = [m/(2M_0)]^(1/3)
Example: globular cluster mass = 1e6 M_sun, Galactic bulge mass 1e10 M_sun ==>
r_J = r_0 (50^0.333)*1e-2 ~ 0.04. Stars beyond 0.04r_0 from the globular cluster are
‘evaporating’.
M31 = Andromeda galaxy
Local Group =
Only 3 spirals
Only 1 elliptical (!)
Lots of dwarf, irregular
galaxies
How did they form?
Z = 1000 = epoch of recombination =
beginning of structure/galaxy/clusters formation
At t~0.3 Myr after Big Bang, the electron opacity of plasma drops a
lot, because of recombination of p and e into H. The universe
becomes transparent, loses radiation pressure support.
Z = 6….5 = epoch of first star formation
Metallicity - age relation for F-stars in the solar neighborhood:
heavy element enrichment with time, but also a great scatter above the lower
envelope...
Closed-box model of heavy element enrichment of ISM/stars
definitions
Def.!
Consider some gas turned into stars, and at the same time
enriched in heavy elements by dying stars
chain differentiation rule
change = input - output of Mh
Like in a closed-box model, metallicity Z is lower in regions
where gas content is lower (LMC, dwarf Irr, or outer radii of M33)
So far so good...
…but let’s compute in our closed-box the mass of stars with metallicity
less than a given value Z
(observations give ~10 times fewer low-Z stars)
LMC = Large Magellanic Cloud, a neighbor bound to the
Milky Way
Rotation
speed
~80 km/s
SMC = Small
Magellanic Cloud
No rotation
Fornax dwarf spheroidal galaxy
Foreground
MW star
Know the differences between: dwarf ellipticals, dwarf spheroidals,
and dwarf irregulars (read Ch.4 of textbook)!