There are closed form solutions but too long and complicated to present here. You can find them in the following books:

"Dynamics of Atmospheric Re-Enty" by Frank J. Regan and Satya M. Anandakrishnan, ISBN 1-56347-048-9, AIAA 1993.

"Atmospheric Reentry, An Introduction to Its Science and Engineering" by John J. Martin, Prentic-Hall, Inc. 1966.

Regan and Anandakrishnan's book is very good but way, way over priced (it in-and-out of print through the AIAA).

John J. Martin's book is a classic and strongly recommended. Martin's book was the foundation for Regan and Anandakrishnan's book.

It's doubtful that you'll find either book in an Australian public library or university research library. I recommend buying John J. Martin's book through the Internet and having it shipped to you.

John J. Martin was/is an interesting guy (if he's still alive, he's very elderly). Martin was a brilliant American engineer doing classified work on ICBM RVs while the technology was originally being developed. Martin wanted to write the first text book on designing ICBM RVs. To get around national security issues, he went on sabatical to England and while there, wrote the book without access to classified documents (Martin was a guest of the brilliant British scientist M.J. Lighthill). That book put Martin's career in the fast lane. He later got a job as associate deputy to the director of the Central Intelligence Agency. No doubt his spook work involved Soviet ICBM technology. Now here's the fun bit: after spending most of his career doing black aerospace work and then employed as a spook, Martin became in 1984 an associate administrator for aeronautics and space technology at NASA headquarters. Go figure that one out! Anyway, his book is a technical masterpiece and not that expensive.

Gary