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I did a backpacking trip once where I was above treeline, there was no running water and I was low on fuel for my stove. While the some of the snow was melting it was draining into the ground so fast there was no liquid water.

What I ended up doing for drinking water was throwing snow on top of a tarp and once it melted gathering the water with a bowl.

Would there be a better way of melting the snow without fire or body heat?

If you have a clean black garbage bag with you (and if you don't, you really should :)), put the snow into the garbage bag, arrange it in a thin layer inside the bag, and lay the bag in the sun on a flat rock (if available), thin layer parallel to the flat rock. Weigh it down with a few rocks to help make contact between the black surface and the snow.

This is essentially your tarp solution, but (a) speeding things up because of the heat absorbing character of the blackness; and (b) avoiding the messiness of scooping because you need only pour. This would also work (albeit slowly) slightly below freezing, assuming sun.

A dark coloured water bottle strapped on top of your pack would absorb quite a lot of solar heat on a sunny day. Getting the snow in would be easier with a wide neck, like a bike bottle or Nalgene. Around freezing point this can be quite effective. If you have flexible clear plastic with you in any form (a large ziplock bag for example) this can be loosely wrapped round the heat collector to act as a greenhouse. Black plastic bags can be used to wrap a clear or white bottle (tightly in a single layer).

Melting snow while moving means your meal stops are shorter, so this is with doing even if fuel is plentiful.

If you're static, a foil blanket can be used to reflect extra sunlight onto the bottle; a 3x increase in heat delivery compared to just a bottle should be easy. Insulate the bottle from the ground in this case.

Once you've got some liquid water in there it probably makes sense to top it up rather than decanting (unless of course you need it immediately). If you're planning on dissolving anything in the water (drink concentrate, soup powder etc) you may as well do so early - it's likely to increase absorption in a clear bottle.

To give an idea of the rate of melting: assume everything stays at 0°C, and you've got a 100% absorbing bike bottle (that's what I've got here to measure). At sea level in full sun that can absorb around 15W. 15W is enough to melt around 2.5 ml/min (half a teaspoonful of water). One bottle isn't enough to provide everything you need even exposed all day (and it won't be this efficient in practice), but it's a worthwhile contribution even in this very basic form. Snow is around 1/4 the density of water, so a bottle loosely filled will melt in around an hour, giving you 1/4 of a bottle of water. Pack it in tight and double all these numbers