The t- is only used in two situations:
- When you would not expect lenition, and the word starts with a vowel
- When you would expect lenition, and the word starts with a lenitable s (not sure if this has been covered yet, but the principle is only use it when you are told to use it).
If you are interested in the history, it used to be there, but it has simply been lost in most situations. Perhaps it will be lost altogether in another 1000 years.