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In the Texas Instruments datasheet for LM555 IC, at Page 11 below Figure 15, the calculations for charging and discharging of the capacitor are given. In those calculations, the capacitor is made to charge halfway only, as concluded by taking time-constant coefficient as ln2. enter image description here

Is there any merit taken in account while considering this, or is it taken arbitrarily?

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The equations can be calculated. In the datasheet based on figure 14 it shows the well known astable mode of operation. In this operation the capacitor charges up to \$\dfrac{2\,V_{cc}}{3}\$ through \$R_A\$ and \$R_B\$ and later discharge through \$R_B\$ until \$\dfrac{V_{cc}}{3}\$, which leads to: \begin{equation} \dfrac{2\,V_{cc}}{3} =V_{cc}+ (\dfrac{V_{cc}}{3} - V_{cc})e^{\frac{-t}{(R_A + R_B)\,C}} \end{equation}

Solving for t we have: \begin{equation} t=ln2\,(R_A+R_B)\,C = 0.693\,(R_A+R_B)\,C \end{equation}

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