what is the legal age that a juvenile can leave home on their own without adult approval.

What you are talking about here is something called "emancipation".  In Georgia, the minimum age that one can be emancipated is it's 16.  Emancipation may occur by operation of law or pursuant to a petition filed by a minor with the juvenile court as provided in this article.

An emancipation occurs by operation of law:

(1) When a minor is validly married;
(2) When a person reaches the age of 18 years; or
(3) During the period when the minor is on active duty with the armed forces of the United States.

An emancipation occurs by court order pursuant to a petition filed by a minor with the juvenile court.  Basically, the minor files a petition in the juvenile court; the parents are then notified; and a hearing is held.  At that time the parents have the opportunity to contest the emancipation.  The court will then order an investigation of the minor’s circumstance, or a ruling on the petition might be made.

It is up to the minor to prove that emancipation is in their best interest and provide a valid reason for the emancipation.  The minor must also must prove that they are capable of supporting themselves without any assistance; this includes such things as paying for rent, utilities, food, medical expenses, etc.  Additionally, the minor must provide the names of adults who have personal knowledge of the minor’s circumstances and believe that emancipation is in the best interest of the minor; these people would include counselors, social workers, teachers, ministers, etc.  The court will then seek an affidavit from each of them.

This is neither an easy or quick process.  And quite frankly, it is rarely ever granted.