We examine the test-particle solution for diffusive shock acceleration , based on simple models for thermal leakage injection and Alfvénic drift .
The critical injection rate , \xi _ { c } , above which the cosmic ray ( CR ) pressure becomes dynamically significant , depends mainly on the sonic shock Mach number , M , and preshock gas temperature , T _ { 1 } .
In the hot-phase interstellar medium ( ISM ) and intracluster medium , \xi _ { c } \lesssim 10 ^ { -3 } for shocks with M \lesssim 5 , while \xi _ { c } \approx 10 ^ { -4 } ( T _ { 1 } / 10 ^ { 6 } { K } ) ^ { 1 / 2 } for shocks with M \gtrsim 10 .
For T _ { 1 } = 10 ^ { 6 } K , for example , the test-particle solution would be valid if the injection momentum , p _ { inj } > 3.8 p _ { th } ( where p _ { th } is thermal momentum ) .
This leads to the postshock CR pressure less than 10 % of the shock ram pressure .
If the Alfvén speed is comparable to the sound speed in the preshock flow , as in the hot-phase ISM , the power-law slope of CR spectrum can be significantly softer than the canonical test-particle slope .
Then the CR spectrum at the shock can be approximated by the revised test-particle power-law with an exponential cutoff at the highest accelerated momentum , p _ { max } ( t ) .
An analytic form of the exponential cutoff is also suggested .