Contents List of Figures

6.3 Fundamental Parameters for Jets 281
The critical density contrast ηcrit can be estimated from the condition that the
bow shock becomes transonic, vHS = cS. For relativistic beams vB c, we find
ηcrit 10 −5 . For η < ηcrit, no bow shock will be found.
For a more quantitative formulation one needs certain empirical relations between
the elliptical’s blue magnitude MB, its soft X–ray emission LX, the stellar
velocity dispersion σ∗ (the Faber–Jackson relation), and the core radius Rc (Kormendy
relation), assuming H0 = 75 (Bicknell 1995)
log LX = 22.3 − 0.872 MB (685)
log σ∗ = 5.412 − 0.0959 MB (686)
log Rc = 11.7 − 0.436 MB . (687)
We furthermore assume a density scaling for the ISM of the elliptical
n(d) =
n0
[1 + (d/Rc) 2 ] δ
(688)
with δ 0.75 typically. Then the hotspot advance speed follows from pressure
balance
vHS(d) = vB
X[1 + (d/Rc) 2 ] δ/2
d/Rc + X[1 + (d/Rc) 2 . (689)
] δ/2
Here X =
4LB/πΘ 2 R 2 cn0µmpc 3 with LB as the beam power, Θ the effective opening
angle for the beam. A reasonable number is Θ 0.1 rad, at least for the inner
jet regions. X corresponds to the above quantity √ χη for cylindrical beams. The
temperature of the X–ray gas is not independent, but tied to the central stellar velocity
dispersion, kT = 2.2µmpσ 2 ∗/δ (Falle 1987). There is evidence that the central
gas density n0 is somewhat higher than the X–ray gas density nX, n0 = κnX with
κ 3. The interstellar density declines quite rapidly at distances beyond a few kpc,
observationally Rc 1 kpc. Hence the most likely regime for the jet’s decollimation
due to hotspots having slowed down to subsonic speeds lies within roughly 10 kpc
of the core.
Gopal–Krishna & Wiita (2001) now evaluate the critical beam power L ∗ B for
which the hotspot deceleration to subsonic velocities occurs at a distance d ∗ 3−10
kpc from the core. 10 kpc is a typical distance at which jets flare in a sample of
radio galaxies. Laing et al. (1999) have studied a sample of 38 FR I sources giving
a mean projected value of 3.5 kpc for the radial distance of the point where the
kpc–scale jet first becomes visible, after passing through an emission gap.
Now X can be scaled to
X = C2
√
LB/ n0Rc
(690)