The same as fathers who are married to the mother. The question is what is in the best interests of the child. Also, your step-daughter can sue for child support and get an enforceable court order. However, support and visitation have nothing to do with each other.
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The law does not differentiate between married fathers and "unwed fathers" in Virginia when it comes to parental rights. The only distinction on this issue is that a child "born of" the marriage is "presumed" to be the child of the husband, which then would need to be rebutted by someone claiming that the husband was not the biological father of the child. Whereas, if a man claimed to be the biological father of a child NOT "born of the marriage" and the other parent disputed paternity, then the father would need to prove paternity by a preponderance of the evidence (this same rule works in reverse if it is the mother asserting that the unwed father is the father and he disputes that fact).
So, if the father now wants contact with his child, he does have legal standing to file a petition for custody and/or visitation with the Juvenile & Domestic Relations District Court in the county where the child resides. The court will then determine based on the evidence presented what custody and visitation arrangements would be in the best interests of this particular child giving due consideration to all of the factors set forth in Va. Code Section 20-124.3. As previous counsel indicated in an earlier response, the mother also has a legal right to petition for child support and the father has a legal duty to provide support to the child, whether he has any involvement with or interest in the child or not.
Facts that you mentioned such as that the father has not previously established any bond or relationship with the child, may be financially unstable (so may not be able to provide a safe and adequate place to "exercise" visitation), and has a hisotry of drug abuse are all factors that would likely weigh against the father having more than limited, possibly supervised visitation at the outset.
Often, in these types of circumstances, the court will start the "stranger" parent off with very short visitations a few times a month to be done in the presence of the other parent or some other person that the child knows and trusts (like a grand-parent). The court would then set a review date (usually around 90 days out) and, if the visitations have been going well and the child is adjusting all right, then the visitations will be expanded and/or unsupervised, etc. This is not a statement of guaranteed or certain outcome, as judges have a great deal of discretion in determining what is in the best interests of the child in any given case -- so, it is possible that one judge might award a regular visitation schedule without conditions and a different judge might deny visitation all together. The outcome described above is simply one of the more common ones that I've seen in my family law practice under similar fact patters.
This response does not create an attorney-client relationship and is intended for general information purposes only.
In Virginia, as a parent he has the right to visitation, unless the Court concludes that it may be detrimental to the child. Having the right to see the child is different from the question of how often he sees the child.