Fermi bubbles: the collimated outburst needed to explain forward-shock
  edges
  The bipolar, nonthermal, high-latitude lobes known as the Fermi bubbles (FBs)
are thought to originate from a massive energy release near the Galactic centre
(GC). We constrain the FB engine and the circumgalactic medium (CGM) by
analytically and numerically modeling the FB edges as strong forward shocks, as
inferred from recent observations. A non-directed energy release produces
shocks too spherical to account for observations even for a maximally massive
Galactic disc, critical CGM rotation, or injection effectively offset from the
GC. In contrast, collimated injection nearly perpendicular to the disc can
account for observations in both ballistic (free expansion) and slowdown
regimes, as we show using a simple stratified evolution model verified by
hydrodynamic simulations. FBs still in their ballistic regime require injection
(at $z\simeq100$ pc heights in our model) with a half-opening angle
$\theta\simeq4^\circ$, a normalized velocity $\beta_{-2}\equiv v/(0.01c)\gtrsim
0.4$, and an energy $E\gtrsim2\beta_{-2}^2\times 10^{55}$ erg, launched
$\mathbb{T}\simeq 3.3\beta_{-2}^{-1}$ Myr ago, showing a distinctive
low-pressure region behind the bubble head. Slowing-down (mass accumulated) FBs
require a faster injection, a thinner jet, a smaller
$E/(\beta_{-2}\theta)^{2}$, and a comparable $\mathbb{T}$, and follow a
ballistic stage that must reach a height $z_{s}\gtrsim 5$ kpc.
