It should be understood that it is FORCE that is required to push an arrow through a deer not energy or momentum. We measure force in POUNDS. Now of course you can't have force without energy but it is still force that is required for any movement. As an example, you have a lot of energy in your muscles but until you use that energy to apply a FORCE to the string of your bow you can not PULL the string back. Just like it takes a FORCE to pull your bow string back it is force that is required to push a broadhead tipped arrow through the deer. How much depends on a lot of things but probably the most important of them to keep in mind is sharpness. A sharp broadhead requires much less force to push through than a dull one. You know this intuitively. Now KE and momentum are values that tell us about how the force can be applied. KE tells us about HOW FAR (distance), momentum tells us for HOW LONG (time). For example 50 foot pounds of KE means that a FORCE of 50 pounds can be applied over a DISTANCE of 1 foot. Or if you'd rather 100 pounds over 1/2 foot. 50 pound seconds of momentum means that a FORCE of 50 pounds can be applied over a TIME of 1 second. Or again if you'd rather 100 pounds for 1/2 second.

So lets look at a real example of an arrow in flight. Let's take a 397.5 grain arrow moving at 238 ft/sec. KE calculates to be 50 ft/lbs and momentum is 0.42 pound seconds. Now lets say that arrow strikes an object that is absolutely uniform in its consitency and given the sharpness of the arrow and frictional resistence and all that complicated stuff it winds up requiring 50 POUNDS of force to push the arrow into the object. How far will it penetrate? The answer is 50 foot pounds divided by 50 pounds = 1 foot. How long will it take? Well if the arrow has 0.42 pound seconds of momentum it will take 0.42 pound seconds divided by 50 pounds = 0.0084 seconds. Notice the literal common denominator is pounds or force.

So KE told us that a 397.5 grain arrow traveling 238 ft/sec will come to a stop in an object that resists the arrows movement with 50 pounds of continuous force in a DISTANCE of 1 foot.
Momentum told us that a 397.5 grain arrow traveling 238 ft/sec will come to a stop in an object that resists the arrows movement with 50 pounds of continuous force in a TIME of 0.0084 seconds.

But keep in mind, FORCE is what is required to make something happen whether it is to draw your string back, launch an arrow or bring an arrow to a stop. NOTHING starts or stops without a FORCE being applied. You can think of KE and momentum as nothing more than measures of how far or for how long can you apply the force.

Now finally to the question. Answer .... It's a trick question! No really, there is no precise answer to this question. There are a whole lot of assumptions that have to be rolled up in any answer before you can even begin. So many, I don't think it even makes sense to start. But think about this. When you pull your string back think about the force and "feel" the force you are using as you draw it back and then think about applying that same force on an arrow "pushing" the arrow with a sharp broadhead on it into a deer. Now think about doubling that force. Minus some efficiency losses (say 20% or so) that will give you a very rough feel for the potential penetrating force you have to push the arrow through the deer.
Did I make any sense whatsoever?
P.S.
[If you're curious about why I said think about "doubling" the force... I used double because the distance through a deer (broadside) is roughly a little less than 1/2 the power stroke on a bow of a typical archer. Sorry but without writing a book on this and boring everyone further, I can't explain it any better. Hope I shed just a little light and mabe gave a little different way of thinking about this and didn't confuse everyone]