Hydrogen Balmer lines are commonly used as spectroscopic effective temperature diagnostics of late-type stars .
However , reliable inferences require accurate model spectra , and the absolute accuracy of classical methods that are based on one-dimensional ( 1D ) hydrostatic model atmospheres and local thermodynamic equilibrium ( LTE ) is still unclear .
To investigate this , we carry out 3D non-LTE calculations for the Balmer lines , performed , for the first time , over an extensive grid of 3D hydrodynamic stagger model atmospheres .
For \mathrm { H \upalpha } , \mathrm { H \upbeta } , and \mathrm { H \upgamma } we find significant 1D non-LTE versus 3D non-LTE differences ( 3D effects ) : the outer wings tend to be stronger in 3D models , particularly for \mathrm { H \upgamma } , while the inner wings can be weaker in 3D models , particularly for \mathrm { H \upalpha } .
For \mathrm { H \upalpha } , we also find significant 3D LTE versus 3D non-LTE differences ( non-LTE effects ) : in warmer stars ( T _ { \mathrm { eff } } \approx 6500 \mathrm { K } ) the inner wings tend to be weaker in non-LTE models , while at lower effective temperatures ( T _ { \mathrm { eff } } \approx 4500 \mathrm { K } ) the inner wings can be stronger in non-LTE models ; the non-LTE effects are more severe at lower metallicities .
We test our 3D non-LTE models against observations of well-studied benchmark stars .
For the Sun , we infer concordant effective temperatures from \mathrm { H \upalpha } , \mathrm { H \upbeta } , and \mathrm { H \upgamma } ; however the value is too low by around 50 \mathrm { K } which could signal residual modelling shortcomings .
For other benchmark stars , our 3D non-LTE models generally reproduce the effective temperatures to within 1 \sigma uncertainties .
For \mathrm { H \upalpha } , the absolute 3D effects and non-LTE effects can separately reach around 100 \mathrm { K } , in terms of inferred effective temperatures .
For metal-poor turn-off stars , 1D LTE models of \mathrm { H \upalpha } can underestimate effective temperatures by around 150 \mathrm { K } .
Our 3D non-LTE model spectra are publicly available , and can be used for more reliable spectroscopic effective temperature determinations .