\begin{table}%ta3
\par
\caption{\label{table:binfracm_powerlaw}The specific binary fraction $\binfm$ for the nine pairing functions described in this paper, resulting from a power-law mass distribution $f_M(M)$ and a uniform mass ratio distribution $h_q(q)$. See Appendices~\ref{appendix:rp}--\ref{appendix:scp} for a detailed description, and Fig.~\ref{figure:binaryfraction_versus_spectraltype} for a visualisation. }
\small%\centering
\par
\begin{tabularx}{\textwidth}{p{0.01cm}XX}
\hline\hline
\par
  \mbox{Pairing function} & \multicolumn{1}{c}{Specific binary fraction} & \multicolumn{1}{c}{Remarks} \\
 \hline
\par
  \mbox{RP} &
\par
\[  
\binfm = \left( \frac{\binf^{-1}-1}{2 F_M(M_1)} + 1 \right)^{-1}
\quad \quad
c \leq M_1 \leq d
\]
\par
&
\par
\[  
F_M(M_1) = \frac{ M_1^{1-\alpha} - c^{1-\alpha} }{ d^{1-\alpha} - c^{1-\alpha} }
\]
\par
\\
\par
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\par
  \hline
\par
  \mbox{PCRP} &
\par
\[  
\binfm = \binf
\quad \quad
c \leq M_1 \leq d
\]
&
\par
\\
\par
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\par
  \hline
\par
  \mbox{PCP-I} &
\par
\[  
\binfm = \binf
\quad \quad
c \leq M_1 \leq d
\]
&
\par
\\
\par
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\par
  \hline
\par
  \mbox{PCP-II} &
\par
\[  
\binfm = \binf \left( 1 - \frac{c}{M_1} \right)
\quad \quad
c \leq M_1 \leq d
\]
&
\par
\\
\par
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\par
  \hline
\par
  \mbox{PCP-III} &
\par
\[  
\binfm = \binf
\quad \quad
c \leq M_1 \leq d
\]
&
\par
\\
\par
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\par
  \hline
\par
  \mbox{SCP-I} &
\par
  \[  
    \binfm = \left\{
    \begin{array}{ll}
      1                                                    & c \leq M_1 \leq 2c \\[5pt]
\displaystyle      \left( 1 + \frac{\alpha+2}{\binf \cdot R(M_1)} \right)^{-1} & 2c \leq M_1 \leq d \\[5pt]
      \displaystyle  \left( 1 + \frac{\alpha+2}{\binf \cdot S(M_1)} \right)^{-1} & d \leq M_1 \leq 2d \\[5pt]
    \end{array}
    \right.
  \]
\par
  &
\par
  \[  
    R(M_1) = 2^{\alpha+2} - \left( 3-\frac{M_1}{c} \right)^{\alpha+2}
  \]
\par
  \[  
    S(M_1) = \left( 3 - \frac{M_1}{d} \right)^{\alpha+2} - 1
  \]
\par
\\
\par
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\par
  \hline
\par
  \mbox{SCP-II} &
\par
  \[  
    \binfm = \left\{
    \begin{array}{ll}
      \binf_{100,R}(M_1)                                                    & c \leq M_1 \leq 2c \\[5pt]
      \displaystyle\frac{\binf_{100,S}(M_1)(\alpha+2)}{1+S(M_1)(\binf^{-1}-1)}           & 2c \leq M_1 \leq d \\[5pt]
     \displaystyle \frac{\binf_{100,T}(M_1)(\alpha+2)}{1+T(M_1)(\binf^{-1}-1)}           & d \leq M_1 \leq 2d-c \\[5pt]
      0                                                                     & 2d-c \leq M_1 \leq 2d \\
    \end{array}
    \right.
  \]
\par
&
\par
  \[  
    \binf_{100,R}(M_1) = 
    \displaystyle\frac{
      2^{\alpha+2} - \left( 1+c/M_1\right)^{\alpha+2}
    }{
      2^{\alpha+2} - \left( 3-M_1/c\right)^{\alpha+2}
    } 
  \]
\par
  \[  
    \binf_{100,S}(M_1) = 
   \displaystyle \frac{
      2^{\alpha+2} - \left( 1+c/M_1\right)^{\alpha+2}
    }{
      2^{\alpha+2} - 1
    } 
  \]
\par
  \[  
    \binf_{100,T}(M_1) = 
   \displaystyle \frac{
      \left( 3-M_1/d\right)^{\alpha+2} - \left( 1+c/M_1\right)^{\alpha+2}
    }{
      \left( 3-M_1/c\right)^{\alpha+2} - 1
    } 
  \]
\par
  \[  
   \displaystyle S(M_1) = 2^{\alpha+2} - \left( 1+\frac{c}{M_1} \right)^{\alpha+2} 
  \]
\par
  \[  
   \displaystyle T(M_1) = \left( 3 - \frac{M_1}{d} \right)^{\alpha+2} - \left( 1+\frac{c}{M_1} \right)^{\alpha+2} 
  \]
\par
\\
\par
  \hline
\par
  \mbox{SCP-III} &
\par
  \[  
    \binfm = \left\{
    \begin{array}{ll}
      1                                                           & c \leq M_1 \leq 2c \\[5pt]
    \displaystyle  \left( 1 + \frac{\alpha+2}{\binf \cdot R(M_1)} \right)^{-1} & 2c \leq M_1 \leq d \\[5pt]
   \displaystyle   \left( 1 + \frac{\alpha+2}{\binf \cdot S(M_1)} \right)^{-1} & d \leq M_1 \leq 2d-c \\[5pt]
      0                                                           & 2d-c \leq M_1 \leq 2d \\
    \end{array}
    \right.
  \]
\par
&
\par
  \[  
  \displaystyle  R(M_1) = \left( 1 - \frac{c}M_1{}  \right)^{-1}
   \displaystyle \left[ 2^{\alpha+2} - \left( 1+\frac{c}{M_1} \right)^{\alpha+2} \right]
  \]
\par
  \[  
 \displaystyle   S(M_1) = \left( 1 - \frac{c}M_1{}  \right)^{-1}
  \displaystyle  \left[ \left( 3 - \frac{M_1}{d} \right)^{\alpha+2} - \left( 1+\frac{c}{M_1} \right)^{\alpha+2} \right]
  \]
\par
\\
\par
  \hline
\par
\end{tabularx}
\par
\end{table}