We measured carbon abundances and the ^ { 12 } \mathrm { C } / ^ { 13 } \mathrm { C } ratio in 31 giant branch stars with previous CN and CH band measurements that span -2.33 < M _ { V } < 0.18 in the globular cluster M10 ( NGC 6254 ) .
Abundances were determined by comparing CO features at \sim 2.3 \mu \mathrm { m } and specifically the ^ { 13 } CO bandhead at 2.37 \mu \mathrm { m } , to synthetic spectra generated with MOOG .
The observed spectra were obtained with GNIRS on Gemini North with a resolution of R \approx 3500 .
The carbon abundances derived from the IR spectra agree with previous [ C/Fe ] measurements found using CN and CH features at the near-UV/blue wavelength range .
We found an average carbon isotope ratio of ^ { 12 } \mathrm { C } / ^ { 13 } \mathrm { C } = 5.10 _ { -0.17 } ^ { +0.18 } for first generation stars ( CN-normal ; 13 stars total ) and ^ { 12 } \mathrm { C } / ^ { 13 } \mathrm { C } = 4.84 _ { -0.22 } ^ { +0.27 } for second generation stars ( CN-enhanced ; 15 stars ) .
We therefore find no statistically significant difference in ^ { 12 } \mathrm { C } / ^ { 13 } \mathrm { C } ratio between stars in either population for the observed magnitude range .
Finally , we created models of the expected carbon , nitrogen , and ^ { 12 } \mathrm { C } / ^ { 13 } \mathrm { C } surface abundance evolution on the red giant branch due to thermohaline mixing using the MESA stellar evolution code .
The efficiency of the thermohaline mixing must be increased to a factor of \approx 60 to match [ C/Fe ] abundances , and by a factor of \approx 666 to match ^ { 12 } \mathrm { C } / ^ { 13 } \mathrm { C } ratios .
We could not simultaneously fit the evolution of both carbon and the ^ { 12 } \mathrm { C } / ^ { 13 } \mathrm { C } ratio with models using the same thermohaline efficiency parameter .