Introduction:

In this insert, we introduce the basics of reference frames relevant to physics.

In particular, we cover inertial frames which are overwhelmingly the MOST relevant reference frames for physics.

Inertial frames are NOT hard to understand: in fact, they are reference frames you use and think about all the time in everyday life and use for Newtonian physics applied to everyday life situations.

But if you want to cover every finicky detail---as yours truly is doomed to do---then the explication goes on and on.

Correctness and clarity are possible, brevity NOT.

What Are Reference Frames?

What are reference frames?

Just a set of coordinates you lay down on geometrical space (e.g., the (3-dimensional physical) space which includes the 3-dimensional space of outer space) or, including time, on spacetime as we say in relativity speak.

See the figure below (local link / general link: frame_reference_spacetime.html) for an example reference frame which includes time coordinates.

Caption: A reference frame laid over spacetime (space and time considered jointly).

Spacetime is a common jargon in relativity speak: i.e., the jargon of relativistic physics (combined special relativity (1905) and general relativity (1915)).

The clocks in the image illustrate the time dimension.

Credit/Permission: User:Maschen, 2012 / Public domain.
Image link: Wikipedia: File:Reference frame and observer.svg.
Local file: local link: frame_reference_spacetime.html.
File: Relativity file: frame_reference_spacetime.html.

Inertial Frames and Physics:

The key feature of inertial frames is that all physical laws in the observable universe are referenced to inertial frames, except general relativity which tells us what inertial frames are.

You can quibble about whether there are other physical laws NOT referenced to inertial frames, but yours truly thinks the quibbling is a matter of perspective or may amount to saying you are NOT using inertial frames in some definitional sense when effectively you are using them.

Fortunately, except in calculations needing general relativity, you can always find an inertial frame to do your calculation in.

In fact, you can find an infinity of inertial frames, and so can choose one that is convenient for your calculation and understanding.

Note:

Our discussion of inertial frames is mostly in the context of Newtonian physics, but there are some necessary features from relativistic physics (combined special relativity (1905) and general relativity (1915)).
Where are inertial frames in general relativity?
You do NOT need to find them. Their properties emerge from general relativity where they are needed. However, full general relativity calculations are usually very computationally expensive and in many cases intractable. So you do NOT use general relativity, except when you need to. And if you are NOT using general relativity, you need to find inertial frames to do your calculations in.
Why do we need multiple inertial frames?
The observable universe as a whole CANNOT be mapped to a single inertial frame. In fact, treating the observable universe as whole requires general relativity. However---and there is always a however---Newtonian physics with the definition of inertial frames given by general relativity (see the below sections Ideal Inertial Frames and Center-of-Mass (CM) Inertial Frames) can describe the observable universe as whole to good approximation.
Isaac Newton (1643--1727) actually posited a theory of a singular fundament inertial frame, but the advent general relativity required that theory to be discarded. For more on Newton's absolute space, see the below section Newton's Absolute Space (Not Required for the RHST).

Ideal Inertial Frames:

Say you have a system of particles and/or objects free falling in a UNIFORM EXTERNAL gravitational field (i.e., one from sources of gravity outside of the system).

NO other EXTERNAL forces act on the system, except the cosmological constant force (or whatever its dark energy equivalent is). The cosmological constant force can actually be considered as an aspect of gravity, and so it usually does NOT need to be mentioned again except when it has a significant effect. We discuss the cosmological constant and cosmological constant force in below section The Cosmological Constant.

The system center of mass (i.e., its mass-weighted average position) can be defined as the origin of an inertial frame.

Three orthogonal axes extending from the origin complete a set of Cartesian coordinates if you choose to use Cartesian coordinates, but you can use any other coordinate system if you like.

The coordinate system is NOT rotating relative to the observable universe (i.e., the bulk mass-energy of the observable universe).

The reference frame specified by the just specified coordinate system is an ideal inertial frame as told to us by general relativity.

Any other coordinate system NOT accelerated relative to the first specified one is also an ideal inertial frame as long as it is still in the region of UNIFORM EXTERNAL gravitational field. So, in fact, you have an infinite set of ideal inertial frames that can be used to analyze the system. You choose the one that is convenient for the analysis.

Any other coordinate system accelerated relative to the first specified one is NOT inertial frame. It is a non-inertial frame and CANNOT be used to analyze the system---UNLESS you convert it to an inertial frame using inertial forces---which you can always do easily in principle, UNLESS you need general relativity which in the classical limit (where Newtonian physics applies), you NEVER do.

Note a reference frame in rotation relative to the observable universe (i.e., a rotating frame) is an accelerated reference frame, and so is non-inertial frame. In fact, every point in a rotating frame has a different acceleration, and so the reference frame is actually a continuum of accelerated reference frames. However, there are inertial forces that will convert rotating frames into inertial frames, and so you can always analyze systems relative to rotating frames and it is often convenient to do so. We discuss rotating frames in the below section Rotating Frames and the Centrifugal Force and the Coriolis Force, Rotating Frames Explicated The Centrifugal Force of the Earth's Rotation, The Shaping of Compact Astro-bodies, The Coriolis Force of the Earth's Rotation and Foucault's Pendulum.

However, as a first word about rotating frames, from our understanding of cosmology, there is an absolute rotation in the observable universe: i.e., rotation relative to the observable universe (i.e., the bulk mass-energy of observable universe). However, we tend NOT to use the expression "absolute rotation" since that seems to have distinct and different meaning (see Wikipedia: Absolute rotation).

In this section, we have been discussing the ideal inertial frames as specified by general relativity However, in general relativity also specifies unideal inertial frames (which include those converted to inertial frames using inertial forces). Ideal and unideal inertial frames are, of course, just called inertial frames.

A general approach to inertial frames is discussed in the below section Center-of-Mass (CM) Inertial Frames.

Center-of-Mass (CM) Inertial Frames:

Yours truly believes that what yours truly calls center-of-mass (CM) inertial frames is a very general specification inertial frames if you need only Newtonian physics plus sometimes a little bit of relativistic physics (combined special relativity (1905) general relativity (1915)).

The procedure for setting up a CM inertial frame for a general system of objects (which system could be a single extended object) is to define the center of mass to be the origin of a coordinate system.

center of mass

mass

average

Wikipedia: Center of mass

You can then analyze the motions of the system relative to the center of mass (i.e., INTERNAL motions) using both INTERNAL forces (i.e., those generated inside system) and EXTERNAL forces (i.e., those generated outside system).

It is usually best to choose the systems for analysis in such away that the EXTERNAL forces are as small as possible.

The center of mass motion relative to some larger EXTERNAL inertial frame that incorporates the CM inertial frame you setting up is determined only by the NET EXTERNAL force (i.e., the INTERNAL forces have NO affect on the center of mass motion) as dictated by Newton's 2nd law of motion (AKA F=ma).

If the system of objects is NOT in acceleration relative the EXTERNAL inertial frame, it is already an inertial frame. If it is in acceleration, you impose inertial forces to convert it to an inertial frame.

Inertial forces in one sense are a trick for converting an non-inertial frame into an inertial frame. However, General relativity tells us there is a fundamental likeness between inertial forces and gravity. In a small closed system where there is NO way of observing the outside world (e.g., a closed room), there is NO way in principle to tell the difference be an UNIFORM gravitational field and an inertial force caused by a constant acceleration. That this is so was, in fact, one of Albert Einstein's (1879--1955) starting point in developing general relativity (1915).

Rather than trying to specify the general procedure further for CM inertial frames (which is tedious and unilluminating), we just consider important special cases in the sections below: Inertial Frames and the Earth through Celestial Frames and Comoving Frames.

Note that in practice people often know immediately what is the appropriate inertial frame to use and do NOT think of general CM inertial frames at all. And also note CM inertial frame is a term yours truly invented since the CM inertial frames needed a name.

Qualification About Rotation Relative to the Observable Universe (Not Required for the RHST):

Actually, a qualification is needed from general relativity about inertial frames and rotation relative to the observable universe. There may be reference frames that are inertial frames (in a sense) rotating relative to the observable universe in very strong gravitational fields such as near black holes. But yours truly CANNOT find any reference that elucidates this qualification. It is hinted at by Wikipedia: Inertial frame of reference: General relativity. Yours truly will usually NOT refer to the qualification again.

Note the qualification is required by general relativity and is NOT present in Newtonian physics as inertial frames are now understood in Newtonian physics.

The Cosmological Constant:

Since the mid 1990s, it has been known that the expansion of the universe is an accelerating expansion of the universe.

The fundamental explanation of the accelerating universe is NOT known. However, the cosmological constant (given symbol capital Greek Lambda = Λ and so often referred to as Lambda Λ) is the simplest explanation and one that gives the right behavior for accelerating universe and structure formation (AKA large-scale structure formation) to good approximation.

In general relativity, the cosmological constant just adds a term to Einstein field equations, and so is, in fact, a feature of gravitation.

In a Newtonian physics sense, the cosmological constant effect acts like as pushing force on matter that grows linearly with distance from any point which means it grows to infinity as you go to infinite distance.

However, the infinity aspect is NOT a concern for calculations interior to a finite system treated with an inertial frame. In such systems, the cosmological constant effect (i.e., the cosmological constant force) just pushes outward from the center of mass on all particles and/or objects

In fact, the cosmological constant force is probably completely negligible for systems smaller than galaxies and probably small even for galaxies and galaxy clusters. Its importance mainly for the comoving frames and the observable universe as whole.

One often hears that accelerating expansion of the universe is caused by dark energy.

In fact, the cosmological constant and dark energy (i.e., constant dark energy) are often conflated because they have the same effect in astrophysical and cosmological contexts. So when someone says the cosmological constant or dark energy, they can mean either one or the other or both. Context decides what they mean.

For simplicity in discussions, yours truly just considers the cosmological constant the simplest form of dark energy even though it is NOT actually a form of energy at all, but an aspect of gravitation as aforesaid.

The constant dark energy is the second simplest form of dark energy (in yours truly's discussions). As aforementioned, dark energy acts just like the cosmological constant in astrophysical and cosmological contexts, but has properties in other contexts.

There are more complicated theories of dark energy called dynamical dark energy theories.

At present, we do NOT know which if any known theory of dark energy (including the cosmological constant) is right.

Dark energy is really a word for our ignorance. We see an effect (i.e., accelerating expansion of the universe) and we attribute it a cause whose properties other than effect we do NOT know.

Inertial Frames and the Earth:

How inertial frames turn up on the Earth in everyday life is explicated in the figure below (local link / general link: frame_inertial_free_fall.html).

Note the discussion in the figure generalizes, mutatis mutandis, to almost all compact astro-bodies (i.e., those NOT held up by kinetic energy, except neutron stars and black holes which require relativistic physics) since they are almost all subject to tidal forces and, since they are almost all in rotation relative to the observable universe, have the rotating frame inertial forces: i.e., the centrifugal force, Coriolis force, and Euler force. We are merely using the Earth as an important-to-us concrete example case.

Caption: The Alien in a free-falling elevator and a free-falling spacecraft in orbit. Both objects define inertial frames. However, they can also be viewed as reference frames in the celestial frame of the Earth since they are accelerated by the gravitational field of the Earth. For more details about reference frames and inertial frames, see Mechanics file frame_basics.html, especially section frame_basics.html: Inertial Frames.

Features:

Both elevator and spacecraft are so small compared to variations in the Earth's gravitational field that they define what are virtually ideal inertial frames.
As for the Earth itself, its center of mass is in free-fall in the external gravitational field of the Solar System. This gravitational field is mainly due to the Sun and Moon.
Of course, the gravitational field of the rest of observable universe is also present, but that is an extremely UNIFORM EXTERNAL gravitational field over the Solar System that it has NO effect on the INTERNAL motions of the Solar System.
In fact, whenever one describes a particular gravitational field as on or over some particular system of astro-bodies, it is always understood that gravitational field of the rest of observable universe is also present, but that gravitational field is an extremely UNIFORM EXTERNAL gravitational field over the particular system, and so has NO effect on the INTERNAL motions of the particular system.
Now say counterfactually that gravitational field of the Solar System over the Earth is exactly UNIFORM and that the Earth is NOT is rotating relative to the observable universe.
The center of mass of the Earth would then be the origin of an ideal inertial frame with coordinate axes attached to the unrotating Earth. The inertial frame can also be described as the center-of-mass (CM) inertial frame of the Earth.
In this counterfactual case, every point on the surface of the Earth is NOT accelerated with respect to the ideal inertial frame attached to the Earth's center of mass so also defines an ideal LOCAL inertial frame (i.e., an inertial frame at the point and nearby surroundings).
You do NOT need for the point to be the center of mass of any object and usually you would NOT call such Earth surface inertial frames CM inertial frames even they were actually CM inertial frames.
For an example of a picturesque piece of Earth surface (but NOT an ideal one), see the figure below (local link / general link: alpine_tundra.html).
Now let's remove one the idealization about the Earth CM inertial frame.
The gravitational field of the Solar System is NOT exactly UNIFORM over the Earth. The Moon's gravitational field and secondarily the Sun's gravitational field vary across the Earth. Note the Moon's gravitational field is weaker than the Sun's, but its variation is greater.
And it is the variation in the gravitational field that causes the tidal force on the Earth (see Mechanics files: tide_earth.html and The Tidal Force and the Earth).
Now the overall gravitational field of the Moon pulls Earth into orbit around the Earth-Moon system center of mass and the gravitational field the Sun pulls the orbit around the Sun (more exactly the Solar System center of mass). .
The tidal forces of the Moon and Sun stretch the Earth along the lines to, respectively, the Moon and Sun.
The stretching causes the Earth tides (i.e., the water tide) and also the Earth's land tide and atmospheric tide.
Now the tidal forces of the Moon and the Sun have virtually NO effect on small-scale everyday life and small-scale laboratory experimentation and so can be neglected for most purposes, but NOT all purposes as discussed below.
For more on the tides, see Mechanics files: tide_earth.html and frame_basics.html: The Tidal Force and Earth.
Now let's remove the second idealization about the Earth CM inertial frame.
The Earth actually has the Earth's daily axial rotation. This means that every point on the surface of the Earth is in acceleration.
Now, in principle, for all calculations you could just use the inertial frame defined Earth's center of mass as an origin with coordinate axes unrotating relative to the observable universe. For brevity, let's call this inertial frame, the UNROTATING FRAME.
In fact, the UNROTATING FRAME is super inconvenient for all purposes since almost all Earthly things (including us) mostly rotate with the Earth.
Therefore, for Earthly purposes, we use a rotating frame that rotates with the Earth and use rotating frame inertial forces to convert said rotating frame in an inertial frame.
The rotating frame inertial forces are the centrifugal force, the Coriolis force, and the Euler force. The Euler force is for accelerating rotation and is NOT needed for the case of the Earth.
We explicate the convertion and the needed rotating frame inertial forces in frame_basics.html: Rotating Frames and the Centrifugal Force and the Coriolis Force.
But the fact is that the acceleration of the surface of the Earth is actually so low (⪅ 0.03 m/s**2: see Wikipedia: Gravity of Earth: Latitude) that for most small-scale purposes (but NOT all purposes), you can treat every surface point as defining a LOCAL inertial frame (i.e., an inertial frame at the point and nearby surroundings) without using the centrifugal force and the Coriolis force.
How small is small scale? Small-scale everyday life and small-scale laboratory experimentation.
When do you need to use the centrifugal force and the Coriolis force? For respectively, the shape of of the Earth and weather.
For more on these uses and rotating frames in general, see Mechanics files: frame_basics.html: Rotating Frames and the Centrifugal Force and the Coriolis Force, frame_basics.html: Rotating Frames Explicated, frame_basics.html: The Centrifugal Force of the Earth's Rotation, and frame_basics.html: The Coriolis Force of the Earth's Rotation.
Note the discussion in this figure generalizes, mutatis mutandis, to almost all compact astro-bodies (i.e., those NOT held up by kinetic energy, except neutron stars and black holes which require relativistic physics) since the overwhelming majority of them are subject to tidal forces and, since they are almost all in rotation relative to the observable universe, and thus are subject to the rotating frame inertial forces: i.e., the centrifugal force, Coriolis force, and Euler force. We are merely using the Earth as an important-to-us concrete example case.

Credit/Permission: © David Jeffery, 2017 / Own work.
Image link: Itself.
Local file: local link: frame_inertial_free_fall.html .
File: Mechanics file: frame_inertial_free_fall.html .

The Tidal Force and the Earth:

First, the tidal force is explicated in the figure below (local link / general link: tidal_force.html).

Caption: A diagram illustrating the tidal force caused by a gravitational field (whose source is off the image to the right) on a free falling spherical astro-body.

The astro-body could be free falling directly to the gravitational field source, but in many actual cases the astro-body is orbiting the center of mass of celestial frame consisting of a system astro-bodies (to which the first astro-body belongs too) and that system (excluding the first astro-body) is collectively the gravitational field source. The simplest case is a gravitationally-bound two-body system.

In brief, the tidal force is a stretching force due to variation in the gravitational field.

Features:

Gravity falls off with distance from the gravitational field source as an inverse-square-law (exactly if the source is spherically symmetric, approximately if NOT) obeying Newton's law of universal gravitation.
Thus, the gravitational force pulls unequally on the astro-body. The nearer parts of the body to the gravitational field source are pulled more strongly than the farther parts. The top panel of the image illustrates this.
If you subtract off the net gravitational force, you have the residual gravitational force trying to stretch the astro-body relative to its center of mass which moves under the net gravitational force only. Note the center of mass motion is NOT effected by INTERNAL forces: they cancel pairwise by Newton's 3rd law of motion in the classical limit (which we are assuming throughout the discussion in this figure).
The residual gravitational force is called the tidal force.
The lower panel of the image illustrates the tidal force and its stretching effect.
Of course, the tidal force just adds to INTERNAL forces acting on the astro-body (e.g., self-gravity, the pressure force, and the centrifugal force) and can be canceled by them usually after a distortion of the astro-body.
If the tidal force gets too strong relative to the INTERNAL forces, the astro-body can be disrupted. This happens to moons that get too close to their host planet.
Also the tidal force can prevent a planetary ring from coalecsing into a moon under its self-gravity.
In general, the tidal force:
1. Increases with the mass of the gravitational field source.
2. Decreases/Increases with increasing/decreasing distance to the gravitational field source.
3. Increases with the size of object acted on by the tidal force.
For a more elaborate explication of the tidal force, see Extended file: Mechanics file: tidal_force_4.html (which could be this file).

Credit/Permission: © William M. Connolley (AKA User:William M. Connolley), 2009 (uploaded to Wikimedia Commons by User::Justass, 2009) / Creative Commons CC BY-SA 3.0.
Image link: Wikipedia: File:Tidal-forces.svg.
Local file: local link: tidal_force.html.
Extended file: Mechanics file: tidal_force_4.html.
File: Mechanics file: tidal_force.html.

Second, the tidal force and Earth are discussed in the figure below (local link / general link: tide_earth.html).

Caption: The tidal forces of the Moon and Sun create the tidal bulges of the World Ocean.

As illustrated, spring tides are the largest tides and neap tides the smallest tides.

Features:

The tidal force is the stretching force due to varying EXTERNAL gravitational field.
Because the Moon is much closer than the Sun, it has a more rapidly varying gravitational field and that turns out to dominate over the overall stronger gravitational field of the Sun.
Hence the tidal force of the Moon is stronger than that of the Sun.
When Moon and Sun are aligned with the Earth (i.e., new moon or full moon), their tidal forces add and tides are largest as aforesaid.
These tides are spring tides where "spring" has the meaning of the jump, burst forth, rise, etc.
Coastal storms that coincide with spring tides are particularly dangerous and prone to cause flooding.
When Moon and Sun are at a right angle as seen from Earth (i.e., first quarter moon or last quarter moon), their tidal forces counteract each other, but Moon's wins, but the tides are smallest as aforesaid.
These tides are neap tides. "Neap is an Old English (AKA Anglo-Saxon) word meaning without the power" (Wikipedia: Range variation: springs and neaps, slightly edited).
As well as the water tide, there is also an Earth tide---but we never notice the continents going up and down by a ∼ 1 meter about twice a day.
There's an atmospheric tide too and furthermore:
See Tide videos below (local link / general link: tide_videos.html).
Tide videos (i.e., Tide videos):
1. NASA: Tidal Forces on Earth | 0:20: An Earth center of mass at rest reference frame is used. Shows how the Earth tides arise. Good for the classroom.
2. Bay of Fundy Tide, Time-lapse, Fundy National Park | 1:18: Bay of Fundy with the world's largest tidal range: 16 meters.
Local file: local link: tide_videos.html.
File: Mechanics file: tide_videos.html.
EOF

local link: tide_earth.html

Mechanics file

tide_earth.html

Rotating Frames and the Centrifugal Force and the Coriolis Force:

Recall that by rotating frames, we mean those rotating relative to the observable universe.

Rotating frames are non-inertial frames, but NOT simple ones.

Every small region in them over a short enough time scale is a simple non-inertial frame (i.e., a reference frame accelerated relative to a local inertial frame) but overall they are a continuum of such simple non-inertial frames.

Nevertheless, they can be converted to inertial frames easily in the classical limit by invoking 3 rotating frame inertial forces: the centrifugal force the Coriolis force, and Euler force. The Euler force is needed for accelerated rotating frames and it is NOT needed for the case of the Earth and we will NOT discuss it further.

The centrifugal force is that "force" that tries to throw you off carnival centrifuges. In the rotating frame, it is an outward pointing body force trying to throw every bit of you outward and an ordinary force has to be exerted on you to hold you in position. From the perspective of the (approximate) inertial frame of the ground, you are just trying to move at a uniform velocity in a straight line per Newton's 1st law of motion.

The Coriolis force is a bit trickier and arises when you have velocity relative to a rotating frame.

Both the centrifugal force and the Coriolis force are important in understanding the internal motions of moons, planets, stars, all other compact astro-bodies (i.e., those NOT held up by kinetic energy) since these are virtually always rotating due to their formation process.

More details on rotating frames are given below in the section Rotating Frames Explicated.

For an important example of the centrifugal force at work, see the below section The Coriolis Force of the Earth's Rotation.

For important example of the Coriolis force at work, see the below section The Coriolis Force of the Earth's Rotation.

Rotating Frames Explicated:

An explication of the basics of rotating frames is given in the figure below (local link / general link: frame_rotating.html).

rotating frame with centrifugal and coriolis forces

Caption: An animation dynamically illustrating rotating frame (attached to a DISK) whose center is at rest relative to a simple EXTERNAL inertial frame (i.e., an EXTERNAL inertial frame NOT rotating relative to the observable universe).

The ball in the animation is sliding frictionlessly on the DISK and NO ORDINARY forces are acting on it at all.

The left-hand panel gives the EXTERNAL inertial frame perspective and the right-hand panel gives the rotating frame perspective.

Below we give an introduction to the basics of rotating frames. Our discussion mostly assumes the classical limit where Newtonian physics applies and relativistic effects are vanishingly small. Occasionally, relativistic effects are mentioned, but a full discussion of them is beyond our scope.

Features:

A simple non-inertial frame is one in uniform acceleration relative to an inertial frame.
The non-inertial frame can be converted into an inertial frame by the introduction of the simple inertial force "-ma" where "m" is the mass of any object under consideration and "a" is the uniform acceleration of the simple non-inertial frame.
Inertial forces are body forces that act equally per unit mass on all bits of a body.
So if a body does NOT resist inertial forces, it suffers NO deformation/strain.
In the classical limit, one can view the introduction of inertial forces as way of generalizing Newton's laws of motion to non-inertial frames.
However, the perspective of the conversion of non-inertial frames to inertial frames seems more useful to yours truly. This is because the conversion to an inertial frame is NOT just a trick. An axiom of general relativity is that almost all physical laws are referenced to inertial frames whether they are simple inertial frames or converted inertial frames. Thus, there is a fundamental likeness of all inertial frames.
General relativity itself is NOT referenced to inertial frames and, in fact, tells us what they are.
One can quibble about whether there are other physical laws NOT referenced to inertial frames, but yours truly thinks the quibbling is a matter of perspective or may amount to saying you are NOT using inertial frames in some definitional sense when effectively you are using them.
A rotating frame (in the sense we mean in this discussion) is one in absolute rotation (i.e., in rotation relative to the observable universe: i.e., to the bulk mass-energy of observable universe) (see Wikipedia: Inertial frame of reference: General relativity).
Now rotating frame is an non-inertial frame, but NOT a simple non-inertial frame (see discussion above). In fact, it is a continuum of simple non-inertial frames each with time-varying acceleration relative to an EXTERNAL inertial frame.
However, by convention, a rotating frame is considered one non-inertial frame. It certainly is one reference frame.
Now there is a formalism to convert a rotating frame into an inertial frame. Below, we explicate this formalism and the animation.
But first note that to avoid tedious and unenlightening generality, we will limit our discussion to rotating frames where the rotation axis does NOT have axial precession relative to the observable universe and is at rest relative to an EXTERNAL inertial frame. We also limit ourselves to when rotating frames have constant angular velocities. These limitations can all be relaxed if one needs to.
The animation conforms to our limitations.
A extreme example of the kind of rotating frame we are NOT discussing is one rotating relative to another rotating frame, but NOT rotating relative to the observable universe. Such tricky cases have their interest, but are finicky to discuss.
In the animation, the left-hand panel shows the ball's motion relative to the EXTERNAL inertial frame. It moves in a straight line at a constant speed, and so is unaccelerated---it's in uniform linear motion. By Newton's 2nd law of motion (AKA F=ma), there is NO ORDINARY net force on the ball as aforesaid.
Recall Newton's laws of motion are referenced to inertial frames although this essential fact is often omitted in high-school presentations.
The right-hand panel shows the ball's motion as seen by an observer in the rotating frame. The ball has an accelerated motion since it follows a curved path despite have NO ORDINARY net force acting on it. But there is NO violation of the Newton's 2nd law since the rotating frame is a non-inertial frame.
But, as aforesaid, we can convert non-inertial frames to inertial frame by introducing inertial forces.
The conversion of a rotating frame (whose center is at rest in an EXTERNAL inertial frame) to an inertial frame is effected by the introduction of 3 rotating frame inertial forces:
1. The centrifugal force which is easy to understand. It's just the force that tries to throw you off carnival centrifuges. From the outside EXTERNAL inertial frame of the ground, you are just trying to move in a straight line at a constant speed in accordance to Newton's 1st law of motion and the force on your back by the carnival centrifuge is needed to accelerate you with the rotating frame of the carnival centrifuge. In general, if you try to stay in one place in the rotating frame, you have to exert real forces to hang on to the rotating frame and stay at rest relative to it.
2. The Coriolis force is trickier to understand. It is a velocity-relative-to-the-rotating frame-dependent force. It depends linearly on the relative relative velocity. If you are NOT moving relative to the rotating frame, the Coriolis force is zero. The Coriolis force is important in weather phenomena in the rotating frame of the Earth's rotation. It's the cause of the vortex motion of cyclones (see Wikipedia: Cyclone: Structure) and anticyclones (see Wikipedia: Anticyclone: Structure). See more explication of cyclones and anticyclones in Mechanics file: coriolis_force.html. The Coriolis force also affects long-range artillery ballistics. We do NOT ordinarily notice the Coriolis force of the Earth's rotation on smaller scales than weather and long-range artillery ballistics. However, on human size scale, Coriolis force causes the behavior of a Foucault pendulum. For an explication of the Foucault pendulum, see Mechanics file: pendulum_foucault.html.
3. The Euler force which arises if the rotation of a rotating frame is accelerating.
The ball in the animation is affected by the centrifugal force and Coriolis force, but NOT by the Euler force since the rotation of the DISK is NOT accelerating.
The main reason the ball follows a curved path in the animation is the Coriolis force since the ball has velocity relative to the rotating frame of the DISK.
All the rotating frame inertial forces arise in playground merry-go-rounds: see the figure below (local link / general link: merry_go_round.html).
For more illustrations of motions in inertial frames and non-inertial frames, see Inertial frames and non-inertial frames videos below (local link / general link: frame_videos.html).
Inertial frames and non-inertial frames videos (i.e., Inertial frames and and non-inertial frames videos):
1. Non-inertial Frames of Reference | 0:47: The ball accelerates (i.e., follows a curved path) even though no horizontal force (other than static friction to cause rolling) because it's in a rotating reference frame which is a non-inertial frame. From the perspective of the approximately inertial frame of the ground, the ball rolls in a straight line just as Newton's 1st law of motion says it should. Both the inertial forces the centrifugal force and the Coriolis force are acting on the ball: the first just by being in a rotating reference frame and the second because of moving relative to the rotating reference frame. Good for the classroom.
2. Top 5 Space Experiments | 10:28: There's a jump to the 4:20 mark. The International Space Station (ISS) is a free-fall frame (i.e., an exact inertial frame), and so Newtonian physics can be referenced directly to the ISS frame and everything behaves completely normally---except for weightlessness of course. Gravity is NOT turned off in space. In low-Earth-orbit, gravity is almost the same as on the Earth's surface, but gravity pulls on everything equally and nothing resists it: everything is in free fall. A few bits are good for the classroom.
3. Guys Play Catch While Skydiving | 0:48: Playing catch the way it should be done. Because of air drag, varying air drag, and probably some level of clear-air turbulence, the system of these 3 bold skydivers is NOT an exact inertial frame, but it's NOT so far from one.
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The Centrifugal Force of the Earth's Rotation:

An explication of how the figure of the Earth is affected by the centrifugal force due to the Earth's rotation is given in the figure below (local link / general link: earth_oblate_spheroid.html).

Caption: "The equatorial (a), polar (b), and mean Earth radii as defined in the 1984 World Geodetic System revision." (Somewhat edited.) These Earth radii roughly specify the figure of the Earth.

The image shows the Western Hemisphere and Americas---somewhat stylized.

Features:

The Earth is actually somewhat oblate: i.e., it is approximately an oblate spheroid (see Wikipedia: Spheroid; Wikipedia: Reference ellipsoid).
Note:
1. Earth Equatorial Radius R_eq_⊕ = 6378.1370 km.
2. Earth polar radius R_po_⊕= 6356.7523 km.
3. R_eq_⊕ - R_po_⊕ = 6378.1370 km - 6356.7523 km = 21.3847 km.
4. Earth mean radius R_me_⊕ = 6371.0088 km by the Earth conventional mean radius formula (2a+b)/3 (see image also), whose rationale is hard to find.
The Earth radii quoted in the last item are for the modern reference ellipsoid for the Earth---which is a more precise specification of the figure of the Earth than just the Earth radii. The reference ellipsoid is a simple geometric shape that approximates the shape of the Earth to high accuracy/precision: to be exact, the shape of an idealized sea level that is continued through the continents. So much for geodesy.
The Earth's atmosphere is probably somewhat oblate too---but that's another story.
The oblateness of the Earth is due to the centrifugal force caused by the Earth's rotation. To expand a bit, the centrifugal force just causes a rearrangement of the bulk Earth so that the Earth's self-gravity, the Earth's pressure force, and the centrifugal force itself balance to create hydrostatic equilibrium everywhere.
The combination of centrifugal force and to some degree the variation in distance from surface to center due to the oblateness causes the effective Earth's gravitational field g (i.e., true Earth's gravitational g_true plus centrifugal force at the Earth's surface) to decrease from the Earth's poles to the equator: 9.832 N/kg to 9.7803267715 N/kg or a decrease of ∼ 0.5 % (see Wikipedia: Gravity of Earth; Wikipedia: Gravity of Earth: Latitude). The variation in effective Earth's gravitational field is below human perception, but quite easily measurable by gravimeters.
Note Earth's gravitational field g_average = 9.80665 N/kg (which is defined in some way) and Earth's gravitational field g_fiducial = 9.8 N/kg (as used by many including yours truly).
The Earth's gravitational field g causes the acceleration due to gravity near the Earth's surface a_⊕, and in fact a_⊕ = g exactly by Newton's 2nd law of motion (AKA F=ma). Note also that the unit N//kg is algebraically equal to the unit m/s**2, but their meaning is different (i.e., they have different quantity dimensions). Note moveover that the actual acceleration in free fall (in the loose everyday sense of falling in air) is affected by air drag.
Note in a strict sense, free fall means a object is moving only under the force of gravity, except for smallish non-gravitational perturbations.
There other causes for the small variations in the Earth's gravitational field: altitude, depth below sea level, and local topography and geology (see Wikipedia: Gravity of Earth: Variation in magnitude).
The Earth Equatorial Radius R_eq_⊕ = 6378.1370 km is the best natural unit distance for astronomy for the Earth-Moon system and near-Earth astronomical objects (natural or artificial) since it is the Earth radius that is naturally used in parallax measurements and is in any case a natural reference natural unit for us Earthlings.
Near-Earth astronomical objects are principally the Moon, Earth-orbiting artificial satellites, space debris, and near Earth objects (NEOs) (asteroids and comets) on close flybys of Earth.

Wikimedia Commons: File:WGS84 mean Earth radius.svg

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The Shaping of Compact Astro-bodies:

Almost all compact astro-bodies (i.e., those NOT held up by kinetic energy, including neutron stars and black holes which require relativistic physics) are subject to the the centrifugal force (since they almost all are rotating relative to the observable universe) and many to the tidal forces (since many are located in relatively compact systems of astro-bodies). These effects cause distortion from the perfect spherical symmetry of hydrostatic equilibrium that the self-gravity and pressure force of an astro-body try to create.

The centrifugal force gives equatorial bulges and the tidal force to tidal bulges which in general will be located in various complex ways.

If both centrifugal force and tidal force are relatively small compared to astro-body's self-gravity and pressure force, they can be treated as perturbations and their effects just added together. If one or both are NOT relatively small, a more detailed treatment is needed.

For the Earth, both centrifugal force and tidal force are relatively small although the centrifugal force has a much bigger effect. The equatorial bulge (evidenced by the fact that the Earth equatorial radius R_eq_⊕ = 6378.1370 km and the Earth polar radius R_po_⊕= 6356.7523 km, and so there is a difference of 21.3847 km) is much larger than the tidal bulges (which for the water tide in the open ocean have a tidal range ∼ 1 m and for the solid Earth tide ∼ 1 m also: see Wikipedia: Tidal_range: Geography Wikipedia: Earth tide: Tidal constituents).

The Coriolis Force of the Earth's Rotation:

An explication of how weather is affected by the Coriolis force due to the Earth's rotation is given in the figure below (local link / general link: coriolis_force.html).

Image 1 Caption: How the Coriolis force (AKA Coriolis effect) due to the Earth's rotation affects weather is illustrated in Image 1.

Features:

The Coriolis force is a rotating frame inertial force that arises from rotation relative to the observable universe (see Mechanics file: frame_rotating.html).
The Coriolis force plus convection are the main cause of band-like atmospheric circulation cells on Earth and other planets: e.g., Jupiter ♃, king of the planets.
Deep atmosphere effects may also play a role in the band structure of gas giant planets.
The Coriolis force is also the cause of cyclones and anticyclones on Earth and other planets. For example, see the following video:
Image 2 Caption: This illustrates how cyclones and anticyclones form in the Northern Hemisphere of Earth.
The Coriolis force causes the clockwise/counterclockwise out/in-spiral of winds from a high-pressure area/low-pressure area in the Northern Hemisphere. The Southern Hemisphere is the mirror image case with the equator being the "mirror".

Image link: Itself.
Image link: Itself.

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Foucault's Pendulum:

The Foucault pendulum and how it demonstrates the Earth's rotation relative to the observable universe is explicated the figure below (local link / general link: pendulum_foucault.html).

Image 1 Caption: An animation of a Foucault pendulum at the North Pole showing the clockwise rotation of its plane of oscillation (i.e., showing its clockwise precession) RELATIVE to the Earth. The animation is time-lapsed: an actual precession period ath the North Pole is a sidereal day = 86164.0905 s = 1 day - 4 m + 4.0905s (on average).

Features:

Inertial frames are NOT in rotation relative to the observable universe---except for rotating frames converted to inertial frames using rotating frame inertial forces (see Mechanics file: frame_rotating.html).
But note that very strong gravitational fields (like those very near black holes) may cause inertial frames to be intrinsically in rotation relative to the observable universe, but this is a tricky point for which yours truly CANNOT find a clear explication. The best so far (and it does NOT say much) is Wikipedia: Inertial frame of reference: General relativity.
A Foucault pendulum illustrates the effect of the rotating frame inertial force the Coriolis force when viewed from the rotating frame of the Earth---which is the opposite viewpoint of the animation in Image 1.
The effect is the precession of the plane of the Foucault pendulum's oscillation relative to the the rotating frame of the Earth.
The precession is caused by the torque of the Coriolis force.
Torque is the twisting manifestation of a force.
Now why does the Coriolis force have this effect for the Foucault pendulum when we do NOT usually see this effect for most pendulums?
The pivot the Foucault pendulum is frictionless, and so the pivot CANNOT exert any torque on the Foucault pendulum. Even a relatively small torque by friction would overcome that of the Coriolis force which is rather weak in this case. Of course, any rigid-direction pivot would completely stop the precession.
Another reason for NOT seeing precession is that even for a Foucault pendulum, the precession period is rather long. The formula for precession period is
```
  P = ( 1 sidereal day )/sin(L) where L is latitude.

    = 1 sidereal day at L = 90°  .

    = [sqrt(3)/2] sdays = (0.8660 ...)  sidereal days at L = 60°  .

    = sqrt(2) sdays = (1.4142 ...) sidereal days at L = 45°  .

    = 2 sidereal days at L = 30°  .

    = ∞ sidereal days at L = 0°  .  
```
(Wikipedia: Foucault pendulum: Examples of precession periods). Note a Foucault pendulum needs some kind of driver to keep it oscillating, but a driver that exerts NO torque.
In the Northern Hemisphere (Southern Hemisphere), the precession clockwise (counterclockwise) relative to the rotating frame of the Earth (Wikipedia: Foucault pendulum: Mechanism).
The simplest locations for a Foucault pendulum are at the poles. There a Foucault pendulum oscillates in a plane fixed relative to the observable universe (which is what is shown in the animation in Image 1) which means the plane of oscillation precesses relative to the Earth once per sidereal day.
The second simplest location is the equator where the plane of oscillation does NOT precess at all relative to the Earth.
The Foucault pendulum is, in fact, a way to demonstrate that Earth surface locations are NOT exactly intrinsic inertial frames on the small size scale.
Large size scale non-inertial frame effects are evidenced by weather (particularly anticyclones and cyclones: see Mechanics file: coriolis_force.html) and long-range artillery ballistics.
Image 2 Caption: An animation of illustrating a Foucault pendulum located in the dome of the Pantheon in Paris. This Foucault pendulum (although maybe NOT Foucault's Foucault pendulum: see below) has length 67 meters and is released at a distance of 50.25 meters (3/4 times its length) in the east with a zero speed. The Foucault pendulum is precessing clockwise RELATIVE to the Pantheon. The precession is time-lapsed to 110 seconds. Alas, the original caption is rather incomplete.
Leon Foucault (1819--1868) used a Foucault pendulum (which he did NOT invent) to demonstrate the Earth's rotation relative to the fixed stars (or as we would say now the observable universe) in 1851 (see Wikipedia: Leon Foucault: Middle years; Wikipedia: Pantheon: Under Louis Philippe I, the Second Republic and Napoleon III (1830-1871)). The Foucault pendulum was a huge one in the dome of the Pantheon in Paris.

Images:

Credit/Permission: Christophe_Dioux, 2023 / CC BY-SA 4.0.
Image link: Wikimedia Commons: File:Foucault pendulum at north pole animated.gif.
Credit/Permission: User:Nbrouard, 2007 / CC BY-SA 3.0.
Image link: Wikimedia Commons: File:Foucault-rotz.gif.

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Celestial Frames and Comoving Frames:

Celestial frames and their largest special case comoving frames are explicated in the the figure below (local link / general link: frame_hierarchy_astro.html.html).

Image 1 Caption: A cartoon illustrating the nested hierarchy of celestial frames (called center-of-mass inertial frames in the cartoon) in the observable universe and an astrophysical rotating frame (attached to, e.g., a moon, planet or star). In our discussion below, we do NOT consider rotating frames. For those, see the Mechanics files: frame_rotating.html and frame_inertial_free_fall.html.

In this figure, we explicate celestial frames and their largest special case comoving frames which are NOT illustrated in the cartoon in Image 1, but of which there is an example in Image 2 below.

Features:

A celestial frame is just a center-of-mass (CM) inertial frames used to analyze systems of astro-bodies. Usually the only ordinary EXTERNAL and INTERNAL force is gravity. Inertial forces sometimes arise, but its beyond our scope to go into that detail.
Note celestial frame is often used as a natural synonym for the system of astro-bodies it is used to analyze.
The EXTERNAL gravitational field determines the center of mass of the system and tidal forces.
Usually, celestial frames are chosen to be gravitationally-bound systems simply because those are ones that tell you most about how the astrophysical realm works. And you can usually analyze a celestial frame with only limited knowledge about its environoment, which it why it is convenient to use them in analysis.
There is a whole nested hierarchy of useful celestial frames all of which are gravitationally-bound systems, except comoving frames. Many celestial frames of one level are nested in one celestial frame of the next level. The center of mass motion of any celestial frame is analyzed in the celestial frame of the next level it is nested in.
The hierarchy is illustrated in Image 1 and is specified (NOT exhaustively) as follows:
1. planet-moon systems: e.g., the Earth-Moon system.
2. planetary systems: e.g., the Solar System.
3. star clusters: Note many planetary systems are NOT in star clusters: e.g., the Solar System.
4. galaxies: e.g., Milky Way.
5. galaxy clusters (e.g., the Virgo Cluster) and galaxy groups (e.g., the Local Group (of Galaxies)). Note a galaxy NOT in a galaxy cluster or galaxy group is a field galaxy.
6. A comoving frame is any spherical region in space large enough to obey the cosmological principle: i.e., its size-scaleless properties (e.g., density, distribution of galaxy types, expansion of the universe behavior) are close to the average of the whole observable universe. The center of mass of the spherical region is its geometrical center since the spherical region is assumed to have nearly UNIFORM density How large does the spherical region have to be? Current thinking is that its size scale has to be ⪆ 370 megaparsecs Mpc (∼ 1000 Mly = 1 Gly) (Wikipedia: Violations of homogeneity; Swala et al. 2025). This size scale can be called the Yadav scale after the lead author of the paper which specified it.
Image 2 Caption: A map of the large-scale structure of the universe of the local universe out to ∼ 300 Mpc = 0.30 Gpc (∼ 2 % of the observable universe radius = 14.3 Gpc) from the center at the unlabeled Milky Way: i.e., to cosmological redshift z ≅ 0.07 and lookback time ≅ 1 Gyr.
The Laniakea Supercluster is marked in yellow in Image 2.
The spherical region in Image 2 is our comoving frame: i.e., the comoving frame centered on us: "us" being any of the Solar System, the Milky Way, and the Local Group. It is of order the Yadav scale: i.e., ⪆ 300 Mpc.
Using measurement of the cosmic microwave background (CMB, T = 2.72548(57) K (Fixsen 2009)), we can, in fact, determine our peculiar velocities relative to the center of mass of our comoving frame (which is where we are in our comoving frame). The center of mass itself just participates in the mean expansion of the universe.
Our peculiar velocities:
1. The Solar System center of mass (i.e., center of mass of the Solar System) is moving at 368(2) km/s in some direction (see Wikipedia: CMB dipole anisotropy (ℓ=1); Caltech: Description of CMB Anisotropies).
2. The Milky Way center of mass is moving at 552(6) km/s in the direction 10.5 hours right ascension (RA), 0.24° declination (Dec or δ) in equatorial coordinates (epoch J2000) which is toward near the center of constellation Hydra (see Wikipedia: Milky Way: Velocity).
3. Local Group of Galaxies center of mass is moving at 627(22) km/s in some direction (see Wikipedia: CMB dipole anisotropy (ℓ=1); Caltech: Description of CMB Anisotropies).
Galaxy superclusters (e.g., those seen in Image 2) are NOT gravitationally bound systems in general. This means they will be pulled apart eventually by the expansion of the universe. Galaxy superclusters are generally NOT analyzed using celestial frames, but in computer simulations of structure formation (AKA large-scale structure formation).

Images:

Credit/Permission: © Richard Powell 2016 (uploaded to Wikipedia by User:AdAstraPerScientiam, 2009) / Creative Commons CC BY-SA 2.5.
Image link: Wikimedia Commons: File:Laniakea.gif.

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Approximate Inertial Frames:

All true theories in physics are only exactly true in ideal limits that can only be approached more or less closely in practice.

Reality just has too many degrees of freedom to absolutely cleanly isolate a theory from all complicating effects, and, of course, NO continuous quantity can ever be measured exactly.

So inertial frames (which follow from a theory in physics) can NEVER be exact.

If an inertial frame is exact enough for your purposes, then you would just refer to it as an inertial frame. If it is NOT exact enough, you could refer to it as an approximate inertial frame.

For example any point on the Earth's surface defines a LOCAL inertial frame that can be just called an inertial frame since it is an exact enough inertial frame for most purposes. It if is NOT quite exact enough for your purposes, you could call it approximate inertial frame. We discuss the inertial frames and the Earth above in section Inertial Frames and the Earth.

Celestial Frames and Newtonian Physics:

Yours truly believes what yours truly calls celestial frames are a good way to understand reference frames used in astrophysics in the classical limit.

The definition of celestial frames is given in Mechanics file: frame_hierarchy_astro.html .

To go beyond the classical limit requires relativistic physics (combined special relativity (1905) general relativity (1915)).

In the classical limit, we use Newtonian physics and certain restrictions apply:

Relative velocities much less than the vacuum light speed c = 2.99792458*10**8 m/s (exact by definition) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns.
Gravitational field much less than near black holes.
Total mass-energy of the astro-bodies much less than that of the observable universe.
Size scale in the analysis much larger than the size scale of atoms and molecules.

The classical limit sounds very restrictive, but, in fact, pretty much everything from cosmic dust to the large scale structure of the universe can be analyzed as in the classical limit in good to excellent approximation depending on the case.

Actually, the observable universe as whole can be treated using Newtonian physics in that the Friedmann equation can be derived from Newtonian physics plus special hypotheses. We will NOT go into that here.

The behavior of black holes close to black holes CANNOT be dealt with by Newtonian physics, but black holes from far enough from them can be treated as point mass sources of gravity by Newtonian physics.

To conclude this section, celestial frames are very general inertial frames for celestial mechanics, but they are NOT completely general. Complete generality is beyond yours truly's scope of knowledge and is probably unnecessary for understanding most reference frames used in astrophysics.

The Fixed Stars and the Observable Universe as Reference Frames:

The fixed stars are traditionally just the relatively nearby stars (e.g., those that historically define the constellations) that are moving in very similar orbits to the Solar System's orbit around the Milky Way.

The fixed stars can be used to approximately measure absolute rotation: i.e., rotation that counts as an accelerated motion relative to an inertial frame. Yours truly prefers to say rotation relative observable universe (i.e., bulk mass of the observable universe) rather than absolute rotation since it is more descriptive of what is actually done to measure absolute rotation using the International Celestial Reference System (ICRS) (see Wikipedia: International Celestial Reference System and its realizations: Realizations).

The fixed stars can be used to approximately measure rotation relative observable universe because their peculiar velocities on average give a rather slow relative to the observable universe.

Historically, the fixed stars were always used until the well into 20th century for measuring absolute rotation at first because this was though to be exactly correct and later because it was the best that could be done. The fixed stars are still used for measuring absolute rotation when you do NOT need the highest accuracy/precision.

Yours truly has almost broken the habit saying "relative to the fixed stars" when yours truly means "relative to the observable universe" which is the exactly correct thing to say when you are discussing absolute rotation.

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Newton's Absolute Space (Not Required for the RHST):

Newton's absolute space was theorized by Isaac Newton (1643--1727) to be the singular fundamental inertial frame (and the one in which the fixed stars [which were all the stars known in his age] were at rest on average) and only reference frames NOT accelerated relative to Newton's absolute space were true inertial frames.

Now yours truly likes the perspective that Newtonian physics is a true emergent theory. It is exactly true in the classical limit.

But NOT Newton's absolute space. That was always a wrong theory.

However, old practitioners of celestial mechanics assuming Newton's absolute space from Newton on until the advent of general relativity in 1915 and even a bit later (see below) still got the right answers for calculations of celestial motions of the systems of astro-bodies they treated (i.e., planet-moon systems, the Solar System, and multiple-star systems). Why?

They treated celestial frames just the way we do, except they could only use the fixed stars for measuring absolute rotation (as we discussed in above section Rotation Relative to the Observable Universe), but that was adequate for their level of accuracy/precision. We still use the fixed stars for measuring absolute rotation, except for the highest level of accuracy/precision for which we use International Celestial Reference System as we discussed in the above section Rotation Relative to the Observable Universe.

So all the calculations of the old practitioners of celestial mechanics got the same answers we do (except at the highest level of accuracy/precision) for the systems of astro-bodies they treated. Newton's absolute space was an adequate theory for their purposes.

However, the old practitioners could NOT have done modern cosmology without the modern understanding of celestial frames since they could NOT have understood the expansion of the universe with Newton's absolute space. Actually, Newton himself partially understood that Newtonian physics was inadequate for cosmology (see No-374--376).

Now Newton and those other old practitioners of celestial mechanics could equally well have anticipated the general relativity perspective of ideal inertial frames (see the above section Ideal Inertial Frames) which is the correct one for the observable universe, but they did NOT do so. If they had, they might have been able to derive some of modern cosmology.

The theory of Newton's absolute space continued to be held by some up to the 1920s. The observational discovery of the expanding universe in 1929 by Edwin Hubble (1889--1953) and its theoretical understanding in terms of the Friedmann-equation (FE) models derived from general relativity caused Newton's absolute space to be thoroughly and most sincerely discarded.

The Basics of Reference Frames Relevant to Physics