You don't even need a birth certificate if you produce secondary evidence that you are an American citizen and it convinces the government that you are.
Secondary Evidence of U.S. Citizenship
Your question is a good segue that Barack Hussein Obama (Soebarkah) is not even a US citizen. Stanley Ann Dunham Soetoro had to cross out Obama's name from her 1968 passport application after an attempt to get him a US passport. She either had a moment of honesty and/or she did not have the proof (or a good scam) to back up her claim that Obama was a US Citizen in 1968 as an addition to her passport application.
The page of the passport application where Stanley Ann Soetoro struck out the name Barack Hussein Obama (Soebarkah)
And a closer look at it.
Here is part of the passport declaration Ann Dunham Soetoro read and heeded.
The passport statement says:
"I have not (and no other person including in the passport or documentation has), since acquiring United States Citizenship, been naturalized as a citizen of a foreign state; take an oath or made an affirmation or other formal declaration of allegiance to a foreign state...."
It is clearly apparent Barack Hussein Obama was not a US citizen by the actions of his own mother, which shows on her passport application. The Stanley Ann Soetoro passport applications came way of a court order to compel the U.S. State Department to comply with a FOIA request.