  The number of satellites in low-Earth orbit is increasing rapidly, and many
tens of thousands of them are expected to be launched in the coming years.
There is a strong concern among the astronomical community about the
contamination of optical and near-infrared observations by satellite trails. We
analyze the impact analysis of such constellations on optical and near-infrared
astronomical observations in a rigorous and quantitative way, using updated
constellation information, and considering imagers and spectrographs and their
very different characteristics. We introduce an analytical method that allows
us to rapidly and accurately evaluate the effect of a very large number of
satellites, accounting for their magnitudes and the effect of trailing of the
satellite image during the exposure. We use this to evaluate the impact on a
series of representative instruments, including imagers (traditional narrow
field instruments, wide-field survey cameras, and astro-photographic cameras)
and spectrographs (long-slit and fibre-fed), taking into account their limiting
magnitude. As already known (Walker et al. 2020), the effect of satellite
trails is more damaging for high-altitude satellites, on wide-field
instruments, or essentially during the first and last hours of the night.
Thanks to their brighter limiting magnitudes, low- and mid-resolution
spectrographs will be less affected, but the contamination will be at about the
same level as that of the science signal, introducing additional challenges.
High-resolution spectrographs will essentially be immune. We propose a series
of mitigating measures, including one that uses the described simulation method
to optimize the scheduling of the observations. We conclude that no single
mitigation measure will solve the problem of satellite trails for all
instruments and all science cases.
