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  • $\begingroup$ Sorry to bother you again John but you said that Carroll gives the potential per unit mass, so to get the potential I need to multiply by $m$ but doing that I get the effective potential as $$ V_{eff}(r) = \frac{1}{2}\epsilon\,m-\frac{GMm}{r} + \frac{m L^2}{2r^2} - \frac{GmML^2}{c^2r^3} \tag{1} $$ If you notice the last two terms are different from the last two term in Eq $(1)$ in you answer. Could you explain this please . $\endgroup$
    – user3741635
    Commented Jun 7, 2016 at 4:59
  • $\begingroup$ Hmm, yes, OK. I'll have to see if I can dig out a copy of Carroll's book to see exactly what he has derived. The form of $V_\text{eff}$ I've given is the one I was taught, and the one that matches Wikipedia in the limit of $M\gg m$. $\endgroup$
    – John Rennie
    Commented Jun 7, 2016 at 5:01
  • $\begingroup$ @user3741635: aha, Carroll derives the expression for a unit mass, i.e. $m=1$, so $m$ does not appear in his equation. $\endgroup$
    – John Rennie
    Commented Jun 7, 2016 at 17:54