We recently discussed arrows vs armour in

https://www.reddit.com/r/AskHistorians/comments/8dhbdx/how_effective_was_a_knights_armor_against_arrows/

from which we can note "To defeat mail: an arrow needs 120J of energy" (from Alan Williams, The Knight and the Blast Furnace, Brill, 2003). A horse archer (or infantry archer) with a powerful bow can deliver that much energy at short range, so mail can be usefully penetrated.

However, note that:

1. Most studies of arrows vs armour have focussed on plate.

2. Plate is simpler to describe - thickness, hardness, and slag content cover the main points - while mail depends on wire thickness, ring diameter, whether it is loose or pulled tight, quality/type of iron, and associated textile armour. There can be a lot of variety in associated textile armour, from no textile armour (just a shirt) through to gambesons both inside and outside the armour.

3. Since mail is flexible, energy that goes into moving the rings isn't available for penetration. This means that details of experimental testing matter a lot - size of test piece, how it is suspended, supported, and backed.

A result of these is that we have much more uncertainty when considering arrows vs mail.

Given that soldiers with mail would, on occasion, wear additional armour over their mail (textile armour, a second layer of mail, a thin breastplate, a coat of plates, lamellar armour), at least some mail without such additional armour was not considered sufficiently protective. Thus, we can conclude that:

• At least some mail armours could be defeated by high draw weight bows.

This is independent of whether the archer is mounted or on foot.

What difference does it make if the archer is mounted?

1. Mounted archers often use less powerful bows (these can still be quite high draw weight - just less so than high draw weight infantry bows).

2. Mounted archers often use efficient reflex-recurve composite bows, which can deliver more energy than a simple longbow of the same draw weight and draw length. This can compensate somewhat for the previous point.

3. The speed of the horse can add to the arrow energy. Since a horse can run at speeds in excess of 10m/s (approximately 30fps), the horse can contribute 10-20% more speed to the arrow, for about 20-50% more energy. Note that this also makes arrows shot at the mounted archer more dangerous!

These mean that we don't expect a large difference between the anti-armour performance of mounted archers and archers on foot.

What about lamellar armour? Lamellar armour often presents 4 thickness to any incoming arrow (otherwise, a combination of 2 and 4 thicknesses). The total thickness of much lamellar armour would be comparable to arrow-proof plate armour. It can be safely assumed that such armour would stop arrows. This is the case for both iron/steel and rawhide lamellar. From Japanese accounts, it was unusual for arrows to penetrate lamellar armour.

Thus:

• All archers with high draw weight bows, whether mounted or not, can endanger armoured soldiers wearing relatively light mail armour without additional armour reinforcing it.

• Lamellar armour, especially when worn as a reinforcement over mail, should stop most or all arrows that hit it.

• Lamellar armour typically has limited coverage, and has gaps - an archer shooting at a soldier armour with lamellar can get lucky. With a powerful bow, they can still be effective even if the gaps are protected by light mail or textile armour.