I live in zone 8b, Northern Arizona.
The winters can get as low as 0F and summer highs 105F.
I want to build an underground greenhouse to moderate the temps. (Want to grow Orange trees inside)
The issue I have is I don't know how much of the greenhouse needs to have direct contact with the earth to take advantage of the deep underground temps.
Is there a ratio of Sunlight-to-Earth so I can gage what the interior temps the greenhouse will be,...Before I build it?

Are you sure you're in zone 8b? I used to live in Flagstaff and it was zoned 5b when I lived there, and is now zoned 6a. For zone 8b the low would be around 20 degrees.
I haven't ever seen anything like you want, but I hope others will know.

Welcome to the Forum: What is your main goal in your posting? Are you trying to save money on purchasing glass/plastic for an underground greenhouse rather than a surface one? You have to have sunlight to grow anything unless you intend to use electricity to grow orange trees. No one does that. Think this through a little. I'm the first guy on the Planet to try anything new, but not convinced on the logic of this one? Keep me informed, perhaps the World is not flat! LOL.

I've seen the type of Gh you're taking about! They are supposed to stay evenly cool/warm within the earth's temperature? IDK a thing about them but the photos I've seen are very cool, like a bunker. They make homes inside of mountains for the same reason. Keep us posted with your progress.

The temperature six to eight feet below the surface stays between 50 and 60F. Do you know your water table? An underground greenhouse should be set at least 5 feet above the water table. The inside can be cased in stone, mud brick or any dense natural material able to absorb large amounts of heat. I don't know if there is any ratio of sunlight to ground, but you are still going to need supplemental heat to grow orange trees if you need temps to be above 50 - 60 F. Granted the greenhouse will probably be a lot hotter during the day when the sun is shining, but on overcast days and at night, they will drop back down. A fossil fuel or solar heat storage system like the one I built for my greenhouse will be needed for these times. Let us know how you proceed. Nice to have you with us in the Greenhouse Forum!

Mike Oehler "The Earth Sheltered Greenhouse", and the Benson Institute has a pdf: on walpinis (or walipinis), http://www.solaripedia.com/fil... that might be helpful. Don't know if this is exactly what you are looking for but might help lead you in the right direction.

Earth is often touted as both mass and insulation an isn't very good as either. Before the availability of closed cell foam and other great performing insulation, a pit greenhouse made a lot of sense compared to a Victorian iron and glass.

Earth sheltered might make sense today if you were building on a south facing slope, but otherwise, not, unless you own a backhoe and enjoy excavating.

The soil temperature isn't stable at 6 feet, it's more like 12 feet here in AZ. I'm near Concho, at 5600 feet. Our soil temp is 63+-2 degrees at 10 foot by my measurement.

Just saying you can easily insulate N, E, W walls to way beyond the performance of cold wet dirt.
It's the glazing losses that will do you in, thermally, and that's the same for pit or above ground.

Pity there aren't more good designs for insulating shutters, curtains, blankets out there.

I read about blown in styrofoam pellets between glass panes for greenhouse insulation. That was before windows sandwiched with gas, so I am unsure if that is better or not. The guy used an old vacuum cleaner for moving the pellets in and out.

Some people are putting rocket stoves in greenhouses and running flues under mass heat storage plant benches. This can make Walipini greenhouses more practical even in cold climates. There is a Calculator online for calculating the correct angle for the glass roof in a Walipini. Sorry I do not know where it is now.