What characterizes recoil? I noticed that people who shoot 9 mm pistols on YouTube experience quite some recoil but that I don't experience any recoil when shooting my compound bow, even though the projectiles of both have similar momentum. Unfortunately, I don't own any firearms and it's been a decade since I shot one, I don't even remember which one, and I didn't ever shoot a pistol (Europe, you know ...), so I can't do a direct comparison.
My compound bow shoots a 27 g arrow at 82 m/s and a 9 mm Commonwealth standard bullet has a mass of 7.5 g. 27 g * 82 m/s / (7.5 g) = 295.2 m/s So it has the same momentum as my arrows when it travels at 295.2 m/s. Its actual speed is 370 m/s, so my arrows have 80% of the momentum of these bullets.
I already almost dismissed the question after I noticed that maybe most of the recoil comes from the exhaust gases, maybe the energy matters, not the momentum, or maybe it's because a bow spends more time accelerating its projectile than a pistol.
But then I went the other way and calculated that a 1 kg metal ball would have to travel at 2.2 m/s to have the same momentum as an arrow. If one were hit by such a ball, the impact would probably be quite noticeable, even when one catches the ball and decelerates it over the distance a bow needs to accelerate an arrow.
So my questions are:


*

*What characterizes recoil? Is it momentum? Is it energy? Is it something else?

*How does a compound bow manage to have no recoil?

 A: *

*Recoil happens due to the conservation of momentum.


In detail:
When the gun is fired, the explosion pushes both the bullet and the gun in opposite directions and, as the initial momentum was zero, the total final momentum must also be zero: so the gun goes backwards with as much momentum as the bullet (and exhaust gases) go forward. The shooter holds the gun, braking its backward movement (and finally transferring this momentum to Earth via friction). Braking the gun requires applying force on it: the reaction to this force, exerted by the gun over the hand, is the recoil force.


*

*In a bow, the force needed to ensure the conservation of momentum is already present before the arrow is shot.


In detail:
The arm holding the bow is already exerting the force necessary to equilibrate the elastic force which, once released, propels the arrow. So when the arrow is shot, the bow exerts a decreasing force on the arrow, and the arm holding the weapon actually has to exert also a decreasing force on the bow, a sort of small "reverse recoil".
Considering, for simplification, a spring device instead of a bow (non-compound, I comment on the general case below), schematically we have, before and after the arrow is shot:

where $\bf{F}_h$ is the horizontal force exerted by the arm holding the bow, $\bf{F}_d$ the force drawing the bow (pulling it back), and $\bf{F}_e$ (shown below) the force from the spring.
Let's consider the forces when the bow is fully drawn:

The force that will propel the arrow ($\bf{F}_e$) is already being exerted by the bow, and being balanced by the archer's drawing force $\bf{F}_d$. The arm holding the bow exerts only a reaction force and, as can be seen from the free body diagrams, $F_h=F_e=F_d$.
Once the spring is released ($\bf{F}_d=0$), the spring exerts a decreasing force on the arrow and an equal (and equaling decreasing) force on the bow. That's why there's no recoil: shooting the arrow doesn't require in any moment the arm holding the bow to exert increased force, on the contrary.
Considering now a compound bow, where a system of pulleys allows $F_d<F_e$, or even a crossbow, which, after cocked, allows $F_d=0$, the recoil can remain negligible, but only because the machine now has heavy moving parts, so the recoil depends on the specific construction. Particularly important is the way the limbs move, if, e.g., they move only sideways (orthogonal to the direction of shooting), the bow receives as much momentum backwards as it imparts to the arrow forward and, just like with a gun, the shooter wasn't already exerting the force to brake the bow, so there'll be a noticeable recoil in this case.
A: Very briefly - momentum is the result of force applied over time. A pistol applies a lot of force over a very short time - in a bow, a smaller force is applied for a much longer time.
The result is that you feel less recoil even though the momentum might be similar. F$\delta t$ is the same - but if $\delta t$ is smaller , F must be larger.
A: There are a few factors playing in to this, and not all of them physics.
Firstly the physics. The momentum of the projectile is (in the case of constant acceleration) force x time. The force is what your experience as recoil. The time over which the projectile is accelerated is very different between a bullet and a bow. For a bullet it is going to be essentially the length of the barrel divided by the speed (give or take a factor of two for the average rather than final speed, but since we are fudging things and pretending the acceleration is constant when it isn't, we're already wrong, and being over precise when making a very approximate calculation is pointless). For the bow it is the length the arrow is drawn back from the neutral position divided by the arrow speed. 
The gun gives you a shorter time by both measures: the barrel length is shorter than the draw of the bow, and the final speed is much higher (by a factor of 4, using your numbers).
Additionally guns have an additional factor of the exhaust gases providing recoil to the gun but not energy to the bullet, so the momentum of the bullet tells you about the lower limit of the recoil, while a bow had no such mechanism: any inefficiencies are turned in to heat and vibration in the bow, not additional backwards momentum of the bow.
There may also be a physiological issue. Holding a gun is more or less a rest state. Holding a drawn bow, the muscles are already under tension. How the body experiences the recoil force is likely to be quite different. In a bow the arm holding the bow is pushing forward with considerable force, already braced against the recoil. With the gun the muscles that are going to oppose the recoil can be completely relaxed until the recoil impulse arrives.
