The majority of galaxy mergers are expected to be minor mergers. The observational signatures of minor mergers are not well understood, thus there exist few constraints on the minor merger rate. This paper seeks to address this gap in our understanding by determining if and when minor mergers exhibit disturbed morphologies and how they differ from the morphology of major mergers. We simulate a series of unequal-mass moderate gas-fraction disc galaxy mergers. With the resulting g-band images, we determine how the time-scale for identifying galaxy mergers via projected separation and quantitative morphology (the Gini coefficient G, asymmetry A, and the second-order moment of the brightest 20% of the light M20) depends on the merger mass ratio, relative orientations and orbital parameters. We find that G-M20 is as sensitive to 9:1 baryonic mass ratio mergers as 1:1 mergers, with observability time-scales ~ 0.2-0.4 Gyr. In contrast, asymmetry finds mergers with baryonic mass ratios between 4:1 and 1:1 (assuming local disc galaxy gas-fractions). Asymmetry time-scales for moderate gas-fraction major disc mergers are ~ 0.2-0.4 Gyr, and less than 0.06 Gyr for moderate gas-fraction minor mergers. The relative orientations and orbits have little effect on the time-scales for morphological disturbances. Observational studies of close pairs often select major mergers by choosing paired galaxies with similar luminosities and/or stellar masses. Therefore, the various ways of finding galaxy mergers (G-M20, A, close pairs) are sensitive to galaxy mergers of different mass ratios. By comparing the frequency of mergers selected by different techniques, one may place empirical constraints on the major and minor galaxy merger rates.
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