Radiant Heating in the Ceiling

I moved into a house from the 50s w/radiant heating in the ceilings. Not sure if the heating is electric or water running through copper pipes (original owner was a plumbing professor, so I’m assuming the latter if such a thing exists). Ceilings are, I believe, plaster and/or lathe. I experienced drops of water dripping from the ceiling in one room today. Not sure if it's from condensation (possible since it's 90 degrees and very humid today) or if we have a leaking pipe. The area w/the drops of water has experienced deterioration (cracked plaster falling off the ceiling and the lathe layer visible) in the last few months. However, today is the first day water has been seen dripping from the ceiling. When touched, the lathe is coming down like sand. I don't know where to start to get this problem fixed. Do I call a plumber since I may be dealing w/copper pipes running through the ceiling? Do I call an HVAC person? My biggest fear is that there is no one out there anymore (they've either retired or are no longer on earth) w/knowledge of and/or experience with our type of heating and we'll be stuck w/having to replace the entire system. Any suggestions and/or info about what we're dealing w/and how to fix it would be greatly appreciated. If anyone has experienced this problem, would love to hear how you dealt w/it too. Thanks.

Mustered up the courage to dig (literally) deeper into this. The radiant heating system is indeed copper pipe. As previously mentioned, the material around the pipe came down like sand as I poked my finger into it to get closer to the leak source. So, I'll move forward first by calling in a plumber. However, I'm now curious about the stuff, which looks like sand, falling from the ceiling. I'm more concerned about its ingredients. I've been reading some of the posts related to radiant heating in ceilings and noticed comments about asbestos. Since asbestos was a common building material at the time my house was built (1950s), I'm wondering if the stuff falling from the ceiling may contain asbestos. My husband called me paranoid as I hunted from hardware store to hardware store looking for a testing kit for asbestos. W/two young children, I'd rather be safe (or as my husband calls me "paranoid") than sorry. Can anyone shed some light on what this material may be? Thanks.

Most likely it's plaster, but may have some asbestos in it.
Put some in a zip lock baggie, and take it to your area health dept, they can point you in the direction you need.