KHHill91: Hmm. I did not know you could mine for shyneum. I remember always being able to buy shyneum at pretty much any trade outpost.
About your initial question: Yes, there is a way. Do you want a spoiler, a broad hint, or do you want to figure it out on your own? You can also find the cluebook online and it will tell you everything you need to know. It is at starflt.com.
NatureDruid: A hint would be nice. Also, where can I sell the ore I find? The starport won't buy it.
Here is your hint: Look at who is affected and who is not affected - What do they have in common? Once you piece this together, it is a pretty easy fix.
Regarding ore, you have to sell it at a trading outpost, but they may or may not want it, and take up cargo space waiting for a buyer, very similar to standard trade goods.
So the simplest way to make money is to go Science Officer and Culture and look up what critters they are buying. At least one of those critters is on the planet. Apparently they are to lazy to hunt for themselves. It helps if there is a very valuable critter. The back of the manual describes the critters and gives a standard trade value.
A tour of the Tandelou worlds will get you enough money for Class 5 Engines in very little time.
The next simplest is to go touring system to system and look for habitable worlds. While you are at it, you can build your information on what worlds trade what special goods. This brings up the real way to make money:
Trade in specialty goods. Find out who makes them and who buys them and you will profit quickly. Sometimes the aliens you meet in space encounters will give you clues.
I would guess you know this already and once you have fully upgraded your ship, there is not much use for money, except to buy fuel. At the trade depot you just go to the "buy" tab and "purchase shyneum" should be right there. If you still have not located it, then I can go take a picture and show you. (EDIT: Added)