Is it permissible for a judge to rule that a father may visit his children one day a week? Is this sufficient to satisfy his affection? Is this ruling from the Sunnah of the Prophet or a command from Allah?

The father has the right to visit his child, who is in the mother's custody, by agreement of the scholars. It is not permissible to prevent either parent from seeing and visiting the child. There is no specific text determining the duration of the visit; rather, the welfare of the child in custody and visitation in a reasonable manner are taken into consideration. If the Sharia court rules for a specific duration, the default assumption is that it ruled based on the child's welfare. The father can appeal the ruling if he believes it to be unjust.