Broad-band photometric colors and effective temperature calibrations for late-type giants -: I.: Z=0.02

We present new synthetic broad-band photometric colors for late-type giants based on synthetic spectra calculated with the PHOENIX model atmosphere code. The grid covers effective temperatures T-eff = 3000... 5000 K, gravities log g = - 0.5... + 3.5, and metallicities [M/H] = + 0.5... - 4.0. We show that individual broad-band photometric colors are strongly affected by model parameters such as molecular opacities, gravity, microturbulent velocity, and stellar mass. Our exploratory 3D modeling of a prototypical late-type giant shows that convection has a noticeable effect on the photometric colors too, as it alters significantly both the vertical and horizontal thermal structures in the outer atmosphere. The differences between colors calculated with full 3D hydrodynamical and 1D model atmospheres are significant (e.g.,. (V- K) similar to 0.2 mag), translating into offsets in effective temperature of up to similar to 70 K. For a sample of 74 late-type giants in the Solar neighborhood, with interferometric effective temperatures and broad-band photometry available in the literature, we compare observed colors with a new PHOENIX grid of synthetic photometric colors, as well as with photometric colors calculated with the MARCS and ATLAS model atmosphere codes. We find good agreement of the new synthetic colors with observations and published T-eff-color and color - color relations, especially in the T-eff ( V - K), T-eff-(J - K) and (J - K) -(V - K) planes. Deviations from the observed trends in the T-eff-color planes are generally within +/- 100 K for T-eff = 3500 to 4800 K. Synthetic colors calculated with different stellar atmosphere models agree to +/- 100 K, within a large range of effective temperatures and gravities. The comparison of the observed and synthetic spectra of late-type giants shows that discrepancies result from the differences both in the strengths of various spectral lines/bands ( especially those of molecular bands, such as TiO, H2O, CO) and the continuum level. Finally, we derive several new T-eff - log g - color relations for late-type giants at solar-metallicity (valid for T-eff = 3500 to 4800 K), based both on the observed effective temperatures and colors of the nearby giants, and synthetic colors produced with PHOENIX, MARCS and ATLAS model atmospheres.