How can Earth-Centered Inertial (ECI) coordinates be properly called "inertial" if the Earth's orbit is always accelerating towards the Sun?

Question

I've just realized (again) that I don't understand anything, something that often happens to me after reading a @DavidHammen answer.

How can Earth-Centered Inertial (ECI) coordinates be inertial if Earth's orbital motion is always accelerating?

$$\mathbf{a} = \frac{d \mathbf{v}}{dt} \approx -\frac{GM_{Sun}}{|\mathbf{r}|^2}$$

has a magnitude of about 0.006 m/s^2 and always points roughly towards the Sun, so we are always falling towards the Sun. We're also accelerating around the Earth-Moon barycenter, and then there's Venus and Jupiter, etc...

Question: How then could ECI be properly called "inertial" if the Earth's center is always accelerating towards the Sun?

Wikipedia sez:

All inertial frames are in a state of constant, rectilinear motion with respect to one another; an accelerometer moving with any of them would detect zero acceleration.

and to me those two sentences don't even match. Yes, in a falling elevator an accelerometer will read zero, but it is not necessarily in "constant, rectilinear motion with respect to" another inertial coordinate system.

Answer 1

How can Earth-Centered Inertial (ECI) coordinates be inertial if Earth's orbital motion is always accelerating?

It is true that "Earth-Centered Inertial" is a bit of a misnomer. What this means is that one has to account for the fictitious acceleration that results from the acceleration of the frame of reference. Unlike the fictitious accelerations that result from working in a rotating frame of reference that all have names (centrifugal acceleration, Coriolis acceleration, and Euler acceleration) the fictitious acceleration that results from using an accelerating frame of reference has no name.

Fortunately, there is a name for the combined effect of this fictitious acceleration due to gravitation of some central body toward a third body and the gravitational acceleration of some object toward the same third body. It is called the "third body effect".

Imagine a satellite in low Earth orbit. The gravitational acceleration of that satellite toward the Sun is nearly the same as is the gravitational acceleration of the Earth toward the Sun. The perturbing effect of the Sun on the satellite's orbit is very small, about four orders of magnitude smaller than the $6\times10^{-3} \text{m/s}^2$ acceleration of the Earth toward the Sun.

This tiny acceleration must be taken into account if one wants to accurately model the satellite's orbit for a long time. But this tiny acceleration can be ignored if high precision isn't needed. Another way of saying this is that an ECI frame is approximately inertial.