In many practical applications, one can consider the Earth-Centered Inertial (ECI) reference frame approximately as an inertial reference system, though strictly speaking, it is non-inertial.
Is there any quotable reference, where this claim is supported by a detailed estimation how small the effects are that one neglects if one considers the Earth's frame as inertial?
Best Answer
The only forces that have a significant impact on the motion of Earth's center-of-mass are gravitational forces. The Earth is "freely falling"; in the language of general relativity, the modern theory of gravity, it is moving along a geodesic.
Because of this fact, it is automatically guaranteed that the spacetime in the vicinity of the Earth's world line is flat to first order; the metric and its first derivatives vanish. (The first derivatives are equivalent to the Christoffel symbol which therefore vanishes at the Earth's center, too.)
This is only modified at the second order: the spacetime curvature (the Riemann tensor) is nonzero near the Earth. Equivalently, the spacetime curvature prevents us from setting the metric tensor equal to the flat spacetime metric at the second order. We may have the metric schematically of the form $$ g_{\mu\nu} (\vec x) \sim \eta_{\mu \nu} + [R_{\alpha\beta\gamma\delta}] [x^\pi x^\rho] $$ So the metric is flat up to corrections that go like $x^2$ where $x$ is the deviation from the Earth's center. These corrections generically manifest themselves as tidal forces; the greatest contributions come from the Moon and the Sun; other planets may matter, too. The non-inertiality of the Solar System as a whole; and the non-inertiality of our local cluster etc. gives increasingly negligible contributions because the tidal forces decrease with the typical distance scale faster than the force itself.
All other deviations from the flat metric are smaller than that. In other words, the tidal forces are the greatest error that you get if you assume that the Earth is an inertial system floating in an empty space; everything else is smaller.
In the text above, I assumed that you use the non-rotating frame of Earth, one that has a fixed orientation relatively to the stars. You may also use a rotating frame that is fixed relatively to the spinning Earth's surface. The inertiality of this rotating frame is obviously violated by the centrifugal and Coriolis forces (and relativistic corrections to them, including frame-dragging etc.).