{ \bf{ Purpose}:} In this paper, the proton resonance width of the near-threshold resonance is calculated using two different approaches. The values of the proton width are used to calculate the low-energy $S$-factor. \\ { \bf{ Method:}} First, the proton resonance width is estimated using the mirror symmetry of the resonance $ { }^{ 11}{ \rm C}(E_{ x}=8.70\, { \rm MeV};\,\frac{ 5}{ 2}^{ +})$ and the mirror bound state ${ }^{ 11}{ \rm B}(E_{ x}=9.272\, { \rm MeV};\,\frac{ 5}{ 2}^{ +})$. In the second approach, this width is estimated using the $R$-matrix definition of the observable resonance width, which is expressed in terms of the $p-{ }^{ 10}{ \rm B}$ resonant wave function calculated in the potential approach and the spectroscopic factor. \\ { \bf{ Results}:} Depending on the method chosen, the calculated proton resonance width varies from $\,1.03 \times 10^{ -19}\,$MeV to $\,2.96 \times 10^{ -19}\,$MeV. The role of the near-threshold resonance is determined using fitting of two low-energy $S$-factors from direct measurements [C. Angulo { \it et al.}, Z. Phys. A { \bf 345}, 333 (1993)] and from the indirect Trojan horse method (THM) [ A. Cvetinovi\'c { \it et al.}, Phys. Rev. C { \bf 97}, 065801 (2018)]. Within the framework of the $R$-matrix method using the determined proton resonance widths and the modified THM parameters for six low-lying resonances, the low-energy $S$-factors were calculated and compared with the corresponding experimental $S$-factors. The closest agreement with the data is achieved with the proton resonance widths $1.0 \times 10^{ -19}$ MeV when fitting the $S$-factor from the THM indirect measurements, and $\,1.68 \times 10^{ -19}\,$ MeV and $\,2.5 \times 10^{ -19}\,$ MeV when fitting the $S$-factor from [Angulo { \it et al}, Z. Phys. A { \bf 345}, 333 (1993)].\
Categories: nucl-th