Current measurements of the \gamma -ray Fermi bubbles ( FB ) are based on model-dependent tracers , carry substantial systematic uncertainties , and are at some tension with each other .
We show that gradient filters pick out the FB edges , which are found to smoothly connect to the bipolar X-ray structure emanating from the Galactic center , thus supporting the interpretation of the FBs as a Galactic-scale phenomenon .
The sharp edges facilitate a direct , model-free measurement of the peripheral FB spectrum .
The result is strikingly similar to the full FB-integrated spectrum , softened by a power law of index \eta \simeq ( 0.2 \mbox { - - } 0.3 ) .
This is naturally explained , in both hadronic and leptonic models , if cosmic rays are injected at the edge , and diffuse away preferentially at higher energies E .
The inferred , averaged diffusion function in the ( more plausible ) leptonic model , D ( E ) \simeq 10 ^ { 29.5 } ( E / 10 \mbox { GeV } ) ^ { 0.48 \pm 0.02 } \mbox { cm } ^ { 2 } \mbox { s } ^ { -1 } , is consistent with estimates for Kraichnan-like turbulence .
Our results , in particular the minute spatial variations in \eta , indicate that the FB edge is a strong , Mach \gtrsim 5 , forward shock .