Here is the way I analyzed it:
Rsource=Rload*(Vopen-Vpeak)/Vpeak
Rsource=500*(4000-2800)/2800
Rsource=214.3
Jout=0.9
Jtotal=Jout*(Rsource+Rload)/Rload
Jtotal=(214.3+500)/500=1.2857
The basic equation for energy stored in a capacitor:
J=0.5*C*V^2
Solving for C,
C=2*J/(V^2)
C=2*1.2857/(4000^2)
C=161nF
Tau=(Rsource+Rload)*C
Tau=114.3usec
If we define pulse width from as being from 90% to 10% (arbitrary maybe, but fairly common for an exponentially decaying pulse),
Pulse Width=2.2*Tau=251usec (I bypassed the derivation of the factor 2.2, because it is well known).
Admittedly, 0.9 joules will not be delivered during the 251usec, but I think the manufacturers could have used the same thought process to come up with total energy first, and then derived (or measured) pulse width afterwards.
The difference between TheOne's derivation and mine is that he constrained the 0.9 Joules to be delivered within 250usec (with some energy remaining in the cap), while I calculated the capacitance required to deliver 0.9 Joules, then calculated the resultant pulse width.
This has become just an interesting intellectual exercise, and I don't know which method (if either) is correct, nor do I particularly care.
Edit:
After I posted this, I concluded that Jtotal should be:
Jtotal=Jout*(Rsource^2+Rload^2)/(Rload^2)
If this is correct, this makes
C=150nF (instead of 160nF), and
PulseWidth=232usec
Not quite as close as before, but still not too far off.