and Cosmology

3.2 Elliptical Galaxies
anisotropic in velocity space. This corresponds to an
anisotropic pressure – where we recall that the pressure
of a gas is nothing but the momentum transport of gas
particles due to their thermal motions.
Derivation of the Collisional Relaxation Time-Scale.
We consider a star passing by another one, with the impact
parameter b being the minimum distance between
the two. From gravitational deflection, the star attains
a velocity component perpendicular to the incoming
direction of
v (1)
⊥
( )( )
Gm 2b
≈ a Δt ≈ b 2 = 2Gm
v bv
v (i)
⊥
, (3.4)
where a is the acceleration at closest separation and Δt
the “duration of the collision”, estimated as Δt = 2b/v
(see Fig. 3.10). Equation (3.4) can be derived more
rigorously by integrating the perpendicular acceleration
along the orbit. A star undergoes many collisions,
through which the perpendicular velocity components
will accumulate; these form two-dimensional vectors
perpendicular to the original direction. After a time t we
have v ⊥ (t) = ∑ i v(i) ⊥
. The expectation value of this vector
is 〈v ⊥ (t)〉 = ∑ 〈 〉
i v (i)
⊥
= 0 since the directions of the
individual v (i)
⊥
are random. But the mean square velocity
perpendicular to the incoming direction does not vanish,
〈
|v⊥ | 2 (t) 〉 = ∑ 〈 〉
v (i) j)
⊥ · v(
⊥
= ∑ 〈 ∣∣∣v (i)
⊥ ∣ 2〉 ̸= 0 , (3.5)
ij
i
〈 〉
j)
where we set · v(
⊥
= 0 for i ̸= j because the
directions of different collisions are assumed to be uncorrelated.
The velocity v ⊥ performs a so-called random
walk. To compute the sum, we convert it into an integral
where we have to integrate over all collision parameters
b. During time t, all collision partners with impact
Fig. 3.10. Sketch related to the derivation of the dynamical
time-scale
parameters within db of b are located in a cylindrical
shell of volume (2πb db)(vt), so that
〈
|v⊥ | 2 (t) 〉 ∫
∣
= 2π b db v tn ∣v (1)
⊥ ∣ 2
( ) 2Gm 2 ∫ db
= 2π v tn
v
b . (3.6)
The integral cannot be performed from 0 to ∞. Thus,
it has to be cut off at b min and b max and then yields
ln(b max /b min ). Due to the finite size of the stellar distribution,
b max = R is a natural choice. Furthermore, our
approximation which led to (3.4) will certainly break
down if v (1)
⊥
is of the same order of magnitude as v;
hence we choose b min = 2Gm/v 2 . With this, we obtain
b max /b min = Rv 2 /(2Gm). The exact choice of the integration
limits is not important, since b min and b max
appear only logarithmically. Next, using the virial theorem,
|E pot |=2E kin , and thus GM/R = v 2 for a typical
star, we get b max /b min ≈ N. Thus,
〈
|v⊥ | 2 (t) 〉 ( ) 2Gm 2
= 2π v tn ln N . (3.7)
v
We define the relaxation time t relax by 〈 |v ⊥ | 2 (t relax ) 〉 = v 2 ,
i.e., the time after which the perpendicular velocity
roughly equals the infall velocity:
( v
2
t relax = 1
2πnv
= 1
2πnv
) 2
1
2Gm ln N
( M
2Rm
) 2
1
ln N ≈ R v
from which we finally obtain (3.3).
3.2.5 Indicators of a Complex Evolution
N
ln N , (3.8)
The isophotes (that is, the curves of constant surface
brightness) of many of the normal elliptical galaxies
are well approximated by ellipses. These elliptical
isophotes with different surface brightnesses are concentric
to high accuracy, with the deviation of the
isophote’s center from the center of the galaxy being
typically 1% of its extent. However, in many cases
the ellipticity varies with radius, so that the value for
ɛ is not a constant. In addition, many ellipticals show
a so-called isophote twist: the orientation of the semimajor
axis of the isophotes changes with the radius.
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